Set of real numbers
You are encouraged to solve this task according to the task description, using any language you may know.
All real numbers form the uncountable set ℝ. Among its subsets, relatively simple are the convex sets, each expressed as a range between two real numbers a and b where a ≤ b. There are actually four cases for the meaning of "between", depending on open or closed boundary:
- [a, b]: {x | a ≤ x and x ≤ b }
- (a, b): {x | a < x and x < b }
- [a, b): {x | a ≤ x and x < b }
- (a, b]: {x | a < x and x ≤ b }
Note that if a = b, of the four only [a, a] would be non-empty.
Task
- Devise a way to represent any set of real numbers, for the definition of 'any' in the implementation notes below.
- Provide methods for these common set operations (x is a real number; A and B are sets):
- x ∈ A: determine if x is an element of A
- example: 1 is in [1, 2), while 2, 3, ... are not.
- A ∪ B: union of A and B, i.e. {x | x ∈ A or x ∈ B}
- example: [0, 2) ∪ (1, 3) = [0, 3); [0, 1) ∪ (2, 3] = well, [0, 1) ∪ (2, 3]
- A ∩ B: intersection of A and B, i.e. {x | x ∈ A and x ∈ B}
- example: [0, 2) ∩ (1, 3) = (1, 2); [0, 1) ∩ (2, 3] = empty set
- A - B: difference between A and B, also written as A \ B, i.e. {x | x ∈ A and x ∉ B}
- example: [0, 2) − (1, 3) = [0, 1]
- Test your implementation by checking if numbers 0, 1, and 2 are in any of the following sets:
- (0, 1] ∪ [0, 2)
- [0, 2) ∩ (1, 2]
- [0, 3) − (0, 1)
- [0, 3) − [0, 1]
Implementation notes
- 'Any' real set means 'sets that can be expressed as the union of a finite number of convex real sets'. Cantor's set needs not apply.
- Infinities should be handled gracefully; indeterminate numbers (NaN) can be ignored.
- You can use your machine's native real number representation, which is probably IEEE floating point, and assume it's good enough (it usually is).
Optional work
- Create a function to determine if a given set is empty (contains no element).
- Define A = {x | 0 < x < 10 and |sin(π x²)| > 1/2 }, B = {x | 0 < x < 10 and |sin(π x)| > 1/2}, calculate the length of the real axis covered by the set A − B. Note that
|sin(π x)| > 1/2 is the same as n + 1/6 < x < n + 5/6 for all integers n; your program does not need to derive this by itself.
C
Providing an implementation of lambdas would be better, but this should do for now. <lang C>#include <math.h>
- include <stdbool.h>
- include <stdio.h>
- include <stdlib.h>
struct RealSet {
bool(*contains)(struct RealSet*, struct RealSet*, double); struct RealSet *left; struct RealSet *right; double low, high;
};
typedef enum {
CLOSED, LEFT_OPEN, RIGHT_OPEN, BOTH_OPEN,
} RangeType;
double length(struct RealSet *self) {
const double interval = 0.00001; double p = self->low; int count = 0;
if (isinf(self->low) || isinf(self->high)) return -1.0; if (self->high <= self->low) return 0.0;
do { if (self->contains(self, NULL, p)) count++; p += interval; } while (p < self->high); return count * interval;
}
bool empty(struct RealSet *self) {
if (self->low == self->high) { return !self->contains(self, NULL, self->low); } return length(self) == 0.0;
}
static bool contains_closed(struct RealSet *self, struct RealSet *_, double d) {
return self->low <= d && d <= self->high;
}
static bool contains_left_open(struct RealSet *self, struct RealSet *_, double d) {
return self->low < d && d <= self->high;
}
static bool contains_right_open(struct RealSet *self, struct RealSet *_, double d) {
return self->low <= d && d < self->high;
}
static bool contains_both_open(struct RealSet *self, struct RealSet *_, double d) {
return self->low < d && d < self->high;
}
static bool contains_intersect(struct RealSet *self, struct RealSet *_, double d) {
return self->left->contains(self->left, NULL, d) && self->right->contains(self->right, NULL, d);
}
static bool contains_union(struct RealSet *self, struct RealSet *_, double d) {
return self->left->contains(self->left, NULL, d) || self->right->contains(self->right, NULL, d);
}
static bool contains_subtract(struct RealSet *self, struct RealSet *_, double d) {
return self->left->contains(self->left, NULL, d) && !self->right->contains(self->right, NULL, d);
}
struct RealSet* makeSet(double low, double high, RangeType type) {
bool(*contains)(struct RealSet*, struct RealSet*, double); struct RealSet *rs;
switch (type) { case CLOSED: contains = contains_closed; break; case LEFT_OPEN: contains = contains_left_open; break; case RIGHT_OPEN: contains = contains_right_open; break; case BOTH_OPEN: contains = contains_both_open; break; default: return NULL; }
rs = malloc(sizeof(struct RealSet)); rs->contains = contains; rs->left = NULL; rs->right = NULL; rs->low = low; rs->high = high; return rs;
}
struct RealSet* makeIntersect(struct RealSet *left, struct RealSet *right) {
struct RealSet *rs = malloc(sizeof(struct RealSet)); rs->contains = contains_intersect; rs->left = left; rs->right = right; rs->low = fmin(left->low, right->low); rs->high = fmin(left->high, right->high); return rs;
}
struct RealSet* makeUnion(struct RealSet *left, struct RealSet *right) {
struct RealSet *rs = malloc(sizeof(struct RealSet)); rs->contains = contains_union; rs->left = left; rs->right = right; rs->low = fmin(left->low, right->low); rs->high = fmin(left->high, right->high); return rs;
}
struct RealSet* makeSubtract(struct RealSet *left, struct RealSet *right) {
struct RealSet *rs = malloc(sizeof(struct RealSet)); rs->contains = contains_subtract; rs->left = left; rs->right = right; rs->low = left->low; rs->high = left->high; return rs;
}
int main() {
struct RealSet *a = makeSet(0.0, 1.0, LEFT_OPEN); struct RealSet *b = makeSet(0.0, 2.0, RIGHT_OPEN); struct RealSet *c = makeSet(1.0, 2.0, LEFT_OPEN); struct RealSet *d = makeSet(0.0, 3.0, RIGHT_OPEN); struct RealSet *e = makeSet(0.0, 1.0, BOTH_OPEN); struct RealSet *f = makeSet(0.0, 1.0, CLOSED); struct RealSet *g = makeSet(0.0, 0.0, CLOSED); int i;
for (i = 0; i < 3; ++i) { struct RealSet *t;
t = makeUnion(a, b); printf("(0, 1] union [0, 2) contains %d is %d\n", i, t->contains(t, NULL, i)); free(t);
t = makeIntersect(b, c); printf("[0, 2) intersect (1, 2] contains %d is %d\n", i, t->contains(t, NULL, i)); free(t);
t = makeSubtract(d, e); printf("[0, 3) - (0, 1) contains %d is %d\n", i, t->contains(t, NULL, i)); free(t);
t = makeSubtract(d, f); printf("[0, 3) - [0, 1] contains %d is %d\n", i, t->contains(t, NULL, i)); free(t);
printf("\n"); }
printf("[0, 0] is empty %d\n", empty(g));
free(a); free(b); free(c); free(d); free(e); free(f); free(g);
return 0;
}</lang>
- Output:
(0, 1] union [0, 2) contains 0 is 1 [0, 2) intersect (1, 2] contains 0 is 0 [0, 3) - (0, 1) contains 0 is 1 [0, 3) - [0, 1] contains 0 is 0 (0, 1] union [0, 2) contains 1 is 1 [0, 2) intersect (1, 2] contains 1 is 0 [0, 3) - (0, 1) contains 1 is 1 [0, 3) - [0, 1] contains 1 is 0 (0, 1] union [0, 2) contains 2 is 0 [0, 2) intersect (1, 2] contains 2 is 0 [0, 3) - (0, 1) contains 2 is 1 [0, 3) - [0, 1] contains 2 is 1 [0, 0] is empty 0
C#
<lang csharp>using System;
namespace RosettaCode.SetOfRealNumbers {
public class Set<TValue> { public Set(Predicate<TValue> contains) { Contains = contains; }
public Predicate<TValue> Contains { get; private set; }
public Set<TValue> Union(Set<TValue> set) { return new Set<TValue>(value => Contains(value) || set.Contains(value)); }
public Set<TValue> Intersection(Set<TValue> set) { return new Set<TValue>(value => Contains(value) && set.Contains(value)); }
public Set<TValue> Difference(Set<TValue> set) { return new Set<TValue>(value => Contains(value) && !set.Contains(value)); } }
}</lang> Test: <lang csharp>using Microsoft.VisualStudio.TestTools.UnitTesting; using RosettaCode.SetOfRealNumbers;
namespace RosettaCode.SetOfRealNumbersTest {
[TestClass] public class SetTest { [TestMethod] public void TestUnion() { var set = new Set<double>(value => 0d < value && value <= 1d).Union( new Set<double>(value => 0d <= value && value < 2d)); Assert.IsTrue(set.Contains(0d)); Assert.IsTrue(set.Contains(1d)); Assert.IsFalse(set.Contains(2d)); }
[TestMethod] public void TestIntersection() { var set = new Set<double>(value => 0d <= value && value < 2d).Intersection( new Set<double>(value => 1d < value && value <= 2d)); Assert.IsFalse(set.Contains(0d)); Assert.IsFalse(set.Contains(1d)); Assert.IsFalse(set.Contains(2d)); }
[TestMethod] public void TestDifference() { var set = new Set<double>(value => 0d <= value && value < 3d).Difference( new Set<double>(value => 0d < value && value < 1d)); Assert.IsTrue(set.Contains(0d)); Assert.IsTrue(set.Contains(1d)); Assert.IsTrue(set.Contains(2d));
set = new Set<double>(value => 0d <= value && value < 3d).Difference( new Set<double>(value => 0d <= value && value <= 1d)); Assert.IsFalse(set.Contains(0d)); Assert.IsFalse(set.Contains(1d)); Assert.IsTrue(set.Contains(2d)); } }
}</lang>
C++
<lang cpp>#include <cassert>
- include <functional>
- include <iostream>
- define _USE_MATH_DEFINES
- include <math.h>
enum RangeType {
CLOSED, BOTH_OPEN, LEFT_OPEN, RIGHT_OPEN
};
class RealSet { private:
double low, high; double interval = 0.00001; std::function<bool(double)> predicate;
public:
RealSet(double low, double high, const std::function<bool(double)>& predicate) { this->low = low; this->high = high; this->predicate = predicate; }
RealSet(double start, double end, RangeType rangeType) { low = start; high = end;
switch (rangeType) { case CLOSED: predicate = [start, end](double d) { return start <= d && d <= end; }; break; case BOTH_OPEN: predicate = [start, end](double d) { return start < d && d < end; }; break; case LEFT_OPEN: predicate = [start, end](double d) { return start < d && d <= end; }; break; case RIGHT_OPEN: predicate = [start, end](double d) { return start <= d && d < end; }; break; default: assert(!"Unexpected range type encountered."); } }
bool contains(double d) const { return predicate(d); }
RealSet unionSet(const RealSet& rhs) const { double low2 = fmin(low, rhs.low); double high2 = fmax(high, rhs.high); return RealSet( low2, high2, [this, &rhs](double d) { return predicate(d) || rhs.predicate(d); } ); }
RealSet intersect(const RealSet& rhs) const { double low2 = fmin(low, rhs.low); double high2 = fmax(high, rhs.high); return RealSet( low2, high2, [this, &rhs](double d) { return predicate(d) && rhs.predicate(d); } ); }
RealSet subtract(const RealSet& rhs) const { return RealSet( low, high, [this, &rhs](double d) { return predicate(d) && !rhs.predicate(d); } ); }
double length() const { if (isinf(low) || isinf(high)) return -1.0; // error value if (high <= low) return 0.0;
double p = low; int count = 0; do { if (predicate(p)) count++; p += interval; } while (p < high); return count * interval; }
bool empty() const { if (high == low) { return !predicate(low); } return length() == 0.0; }
};
int main() {
using namespace std;
RealSet a(0.0, 1.0, LEFT_OPEN); RealSet b(0.0, 2.0, RIGHT_OPEN); RealSet c(1.0, 2.0, LEFT_OPEN); RealSet d(0.0, 3.0, RIGHT_OPEN); RealSet e(0.0, 1.0, BOTH_OPEN); RealSet f(0.0, 1.0, CLOSED); RealSet g(0.0, 0.0, CLOSED);
for (int i = 0; i <= 2; ++i) { cout << "(0, 1] ∪ [0, 2) contains " << i << " is " << boolalpha << a.unionSet(b).contains(i) << "\n"; cout << "[0, 2) ∩ (1, 2] contains " << i << " is " << boolalpha << b.intersect(c).contains(i) << "\n"; cout << "[0, 3) - (0, 1) contains " << i << " is " << boolalpha << d.subtract(e).contains(i) << "\n"; cout << "[0, 3) - [0, 1] contains " << i << " is " << boolalpha << d.subtract(f).contains(i) << "\n"; cout << endl; }
cout << "[0, 0] is empty is " << boolalpha << g.empty() << "\n"; cout << endl;
RealSet aa( 0.0, 10.0, [](double x) { return (0.0 < x && x < 10.0) && abs(sin(M_PI * x * x)) > 0.5; } ); RealSet bb( 0.0, 10.0, [](double x) { return (0.0 < x && x < 10.0) && abs(sin(M_PI * x)) > 0.5; } ); auto cc = aa.subtract(bb); cout << "Approximate length of A - B is " << cc.length() << endl;
return 0;
}</lang>
- Output:
(0, 1] ? [0, 2) contains 0 is true [0, 2) ? (1, 2] contains 0 is false [0, 3) - (0, 1) contains 0 is true [0, 3) - [0, 1] contains 0 is false (0, 1] ? [0, 2) contains 1 is true [0, 2) ? (1, 2] contains 1 is false [0, 3) - (0, 1) contains 1 is true [0, 3) - [0, 1] contains 1 is false (0, 1] ? [0, 2) contains 2 is false [0, 2) ? (1, 2] contains 2 is false [0, 3) - (0, 1) contains 2 is true [0, 3) - [0, 1] contains 2 is true [0, 0] is empty is false Approximate length of A - B is 2.07587
Clojure
<lang Clojure>(ns rosettacode.real-set)
(defn >=|<= [lo hi] #(<= lo % hi))
(defn >|< [lo hi] #(< lo % hi))
(defn >=|< [lo hi] #(and (<= lo %) (< % hi)))
(defn >|<= [lo hi] #(and (< lo %) (<= % hi)))
(def ⋃ some-fn) (def ⋂ every-pred) (defn ∖
([s1] s1) ([s1 s2] #(and (s1 %) (not (s2 %)))) ([s1 s2 s3] #(and (s1 %) (not (s2 %)) (not (s3 %)))) ([s1 s2 s3 & ss] (fn [x] (every? #(not (% x)) (list* s1 s2 s3 ss)))))
(clojure.pprint/pprint
(map #(map % [0 1 2]) [(⋃ (>|<= 0 1) (>=|< 0 2)) (⋂ (>=|< 0 2) (>|<= 1 2)) (∖ (>=|< 0 3) (>|< 0 1)) (∖ (>=|< 0 3) (>=|<= 0 1))])
(def ∅ (constantly false)) (def R (constantly true)) (def Z integer?) (def Q ratio?) (def I #(∖ R Z Q)) (def N #(∖ Z neg?))</lang>
Common Lisp
Common Lisp has a standard way to represent intervals. <lang lisp>(deftype set== (a b) `(real ,a ,b)) (deftype set<> (a b) `(real (,a) (,b))) (deftype set=> (a b) `(real ,a (,b))) (deftype set<= (a b) `(real (,a) ,b))
(deftype set-union (s1 s2) `(or ,s1 ,s2)) (deftype set-intersection (s1 s2) `(and ,s1 ,s2)) (deftype set-diff (s1 s2) `(and ,s1 (not ,s2)))
(defun in-set-p (x set)
(typep x set))
(defun test ()
(let ((set '(set-union (set<= 0 1) (set=> 0 2)))) (assert (in-set-p 0 set)) (assert (in-set-p 1 set)) (assert (not (in-set-p 2 set)))) (let ((set '(set-intersection (set=> 0 2) (set<= 1 2)))) (assert (not (in-set-p 0 set))) (assert (not (in-set-p 1 set))) (assert (not (in-set-p 2 set)))) (let ((set '(set-diff (set=> 0 3) (set<> 0 1)))) (assert (in-set-p 0 set)) (assert (in-set-p 1 set)) (assert (in-set-p 2 set))) (let ((set '(set-diff (set<= 0 3) (set== 0 1)))) (assert (not (in-set-p 0 set))) (assert (not (in-set-p 1 set))) (assert (in-set-p 2 set))))</lang>
D
<lang d>struct Set(T) {
const pure nothrow bool delegate(in T) contains;
bool opIn_r(in T x) const pure nothrow { return contains(x); }
Set opBinary(string op)(in Set set) const pure nothrow if (op == "+" || op == "-") { static if (op == "+") return Set(x => contains(x) || set.contains(x)); else return Set(x => contains(x) && !set.contains(x)); }
Set intersection(in Set set) const pure nothrow { return Set(x => contains(x) && set.contains(x)); }
}
unittest { // Test union.
alias DSet = Set!double; const s = DSet(x => 0.0 < x && x <= 1.0) + DSet(x => 0.0 <= x && x < 2.0); assert(0.0 in s); assert(1.0 in s); assert(2.0 !in s);
}
unittest { // Test difference.
alias DSet = Set!double; const s1 = DSet(x => 0.0 <= x && x < 3.0) - DSet(x => 0.0 < x && x < 1.0); assert(0.0 in s1); assert(0.5 !in s1); assert(1.0 in s1); assert(2.0 in s1);
const s2 = DSet(x => 0.0 <= x && x < 3.0) - DSet(x => 0.0 <= x && x <= 1.0); assert(0.0 !in s2); assert(1.0 !in s2); assert(2.0 in s2);
const s3 = DSet(x => 0 <= x && x <= double.infinity) - DSet(x => 1.0 <= x && x <= 2.0); assert(0.0 in s3); assert(1.5 !in s3); assert(3.0 in s3);
}
unittest { // Test intersection.
alias DSet = Set!double; const s = DSet(x => 0.0 <= x && x < 2.0).intersection( DSet(x => 1.0 < x && x <= 2.0)); assert(0.0 !in s); assert(1.0 !in s); assert(2.0 !in s);
}
void main() {}</lang>
Delphi
<lang Delphi> program Set_of_real_numbers;
{$APPTYPE CONSOLE}
uses
System.SysUtils;
type
TSet = TFunc<Double, boolean>;
function Union(a, b: TSet): TSet; begin
Result := function(x: double): boolean begin Result := a(x) or b(x); end;
end;
function Inter(a, b: TSet): TSet; begin
Result := function(x: double): boolean begin Result := a(x) and b(x); end;
end;
function Diff(a, b: TSet): TSet; begin
Result := function(x: double): boolean begin Result := a(x) and not b(x); end;
end;
function Open(a, b: double): TSet; begin
Result := function(x: double): boolean begin Result := (a < x) and (x < b); end;
end;
function closed(a, b: double): TSet; begin
Result := function(x: double): boolean begin Result := (a <= x) and (x <= b); end;
end;
function opCl(a, b: double): TSet; begin
Result := function(x: double): boolean begin Result := (a < x) and (x <= b); end;
end;
function clOp(a, b: double): TSet; begin
Result := function(x: double): boolean begin Result := (a <= x) and (x < b); end;
end;
const
BOOLSTR: array[Boolean] of string = ('False', 'True');
begin
var s: TArray<TSet>; SetLength(s, 4);
s[0] := Union(opCl(0, 1), clOp(0, 2)); // (0,1] ? [0,2) s[1] := Inter(clOp(0, 2), opCl(1, 2)); // [0,2) n (1,2] s[2] := Diff(clOp(0, 3), open(0, 1)); // [0,3) - (0,1) s[3] := Diff(clOp(0, 3), closed(0, 1)); // [0,3) - [0,1]
for var i := 0 to High(s) do begin for var x := 0 to 2 do writeln(format('%d e s%d: %s', [x, i, BOOLSTR[s[i](x)]])); writeln; end; readln;
end.</lang>
EchoLisp
Implementation of sets operations, which apply to any subsets of ℜ defined by a predicate.
Sets operations
<lang scheme> (lib 'match) ;; reader-infix macros
(reader-infix '∈ ) (reader-infix '∩ ) (reader-infix '∪ ) (reader-infix '⊖ ) ;; set difference
(define-syntax-rule (∈ x a) (a x)) (define-syntax-rule (∩ a b) (lambda(x) (and ( a x) (b x)))) (define-syntax-rule (∪ a b) (lambda(x) (or ( a x) (b x)))) (define-syntax-rule (⊖ a b) (lambda(x) (and ( a x) (not (b x)))))
- predicates to define common sets
(define (∅ x) #f) ;; the empty set predicate (define (Z x) (integer? x)) (define (N x) (and (Z x) (>= x 0))) (define (Q x) (rational? x)) (define (ℜ x) #t)
- predicates to define convex sets
(define (⟦...⟧ a b)(lambda(x) (and (>= x a) (<= x b)))) (define (⟦...⟦ a b)(lambda(x) (and (>= x a) (< x b)))) (define (⟧...⟧ a b)(lambda(x) (and (> x a) (<= x b)))) (define (⟧...⟦ a b)(lambda(x) (and (> x a) (< x b)))) </lang>
- Output:
(3/7 ∈ ∅) → #f (3/7 ∈ Q) → #t (6.7 ∈ ℜ) → #t (define A (⟦...⟧ 2 10)) ; closed interval (define B (⟧...⟦ 5 15)) ; open interval (8 ∈ A) → #t (11 ∈ A)→ #f (define AB (A ∩ B)) (8 ∈ AB) → #t (3 ∈ AB) → #f (5 ∈ AB) → #f ;; because B is ]5 .. 15] (define A-B (A ⊖ B)) (5 ∈ A-B) → #t (-666 ∈ (⟧...⟧ -Infinity 0 )) → #t ;; task (0 ∈ ((⟧...⟧ 0 1) ∪ (⟦...⟦ 0 2))) → #t (0 ∈ ((⟦...⟦ 0 2) ∩ (⟧...⟧ 1 2))) → #f (0 ∈ ((⟦...⟦ 0 3) ⊖ (⟧...⟦ 0 1))) → #t (0 ∈ ((⟦...⟦ 0 3) ⊖ (⟦...⟧ 0 1))) → #f (1 ∈ ((⟧...⟧ 0 1) ∪ (⟦...⟦ 0 2))) → #t (1 ∈ ((⟦...⟦ 0 2) ∩ (⟧...⟧ 1 2))) → #f (1 ∈ ((⟦...⟦ 0 3) ⊖ (⟧...⟦ 0 1))) → #t (1 ∈ ((⟦...⟦ 0 3) ⊖ (⟦...⟧ 0 1))) → #f (2 ∈ ((⟧...⟧ 0 1) ∪ (⟦...⟦ 0 2))) → #f (2 ∈ ((⟦...⟦ 0 2) ∩ (⟧...⟧ 1 2))) → #f (2 ∈ ((⟦...⟦ 0 3) ⊖ (⟧...⟦ 0 1))) → #t (2 ∈ ((⟦...⟦ 0 3) ⊖ (⟦...⟧ 0 1))) → #t
Optional : measuring sets
<lang scheme>
- The following applies to convex sets ⟧...⟦ Cx,
- and families F of disjoint convex sets.
- Cx are implemented as vectors [lo, hi]
(define-syntax-id _.lo [_ 0]) (define-syntax-id _.hi [_ 1]) ;; Cx-ops (define (Cx-new lo hi) (if (< lo hi) (vector lo hi) Cx-empty)) (define (Cx-empty? A) (>= A.lo A.hi)) (define Cx-empty #(+Infinity -Infinity)) (define (Cx-inter A B) (Cx-new (max A.lo B.lo) (min A.hi B.hi))) (define (Cx-measure A) (if (< A.lo A.hi) (- A.hi A.lo) 0)) ;; Families ops (define (CF-measure FA) (for/sum ((A FA)) (Cx-measure A))) ;; because disjoint ;; intersection of two families (define (CF-inter FA FB) (for*/list ((A FA)(B FB)) (Cx-inter A B))) ;; measure of FA/FB = m(FA) - m (FA ∩ FB) (define (CF-measure-FA/FB FA FB) (- (CF-measure FA) (CF-measure (CF-inter FA FB))))
- Application
- FA = {x | 0 < x < 10 and |sin(π x²)| > 1/2 }
(define FA (for/list ((n 100)) (Cx-new (sqrt (+ n (// 6))) (sqrt (+ n (// 5 6))))))
- FB = {x | 0 < x < 10 and |sin(π x)| > 1/2 }
(define FB (for/list ((n 10)) (Cx-new (+ n (// 6)) (+ n (// 5 6)))))
→ (#(0.1667 0.8333) #(1.1667 1.8333) #(2.1667 2.8333)
#(3.1667 3.8333) #(4.1667 4.8333) #(5.1667 5.8333) #(6.1667 6.8333) #(7.1667 7.8333) #(8.1667 8.8333) #(9.1667 9.8333)) (CF-measure-FA/FB FA FB) → 2.075864841184666
</lang>
Elena
ELENA 4.x : <lang elena>import extensions;
extension setOp {
union(func) = (val => self(val) || func(val) ); intersection(func) = (val => self(val) && func(val) ); difference(func) = (val => self(val) && (func(val)).Inverted );
}
public program() {
// union var set := (x => x >= 0.0r && x <= 1.0r ).union:(x => x >= 0.0r && x < 2.0r ); set(0.0r).assertTrue(); set(1.0r).assertTrue(); set(2.0r).assertFalse(); // intersection var set2 := (x => x >= 0.0r && x < 2.0r ).intersection:(x => x >= 1.0r && x <= 2.0r ); set2(0.0r).assertFalse(); set2(1.0r).assertTrue(); set2(2.0r).assertFalse(); // difference var set3 := (x => x >= 0.0r && x < 3.0r ).difference:(x => x >= 0.0r && x <= 1.0r ); set3(0.0r).assertFalse(); set3(1.0r).assertFalse(); set3(2.0r).assertTrue();
}</lang>
F#
<lang fsharp>open System
let union s1 s2 =
fun x -> (s1 x) || (s2 x);
let difference s1 s2 =
fun x -> (s1 x) && not (s2 x)
let intersection s1 s2 =
fun x -> (s1 x) && (s2 x)
[<EntryPoint>] let main _ =
//test set union let u1 = union (fun x -> 0.0 < x && x <= 1.0) (fun x -> 0.0 <= x && x < 2.0) assert (u1 0.0) assert (u1 1.0) assert (not (u1 2.0))
//test set difference let d1 = difference (fun x -> 0.0 <= x && x < 3.0) (fun x -> 0.0 < x && x < 1.0) assert (d1 0.0) assert (not (d1 0.5)) assert (d1 1.0) assert (d1 2.0)
let d2 = difference (fun x -> 0.0 <= x && x < 3.0) (fun x -> 0.0 <= x && x <= 1.0) assert (not (d2 0.0)) assert (not (d2 1.0)) assert (d2 2.0)
let d3 = difference (fun x -> 0.0 <= x && x <= Double.PositiveInfinity) (fun x -> 1.0 <= x && x <= 2.0) assert (d3 0.0) assert (not (d3 1.5)) assert (d3 3.0)
//test set intersection let i1 = intersection (fun x -> 0.0 <= x && x < 2.0) (fun x -> 1.0 < x && x <= 2.0) assert (not (i1 0.0)) assert (not (i1 1.0)) assert (not (i1 2.0))
0 // return an integer exit code</lang>
Go
Just the non-optional part: <lang go>package main
import "fmt"
type Set func(float64) bool
func Union(a, b Set) Set { return func(x float64) bool { return a(x) || b(x) } } func Inter(a, b Set) Set { return func(x float64) bool { return a(x) && b(x) } } func Diff(a, b Set) Set { return func(x float64) bool { return a(x) && !b(x) } } func open(a, b float64) Set { return func(x float64) bool { return a < x && x < b } } func closed(a, b float64) Set { return func(x float64) bool { return a <= x && x <= b } } func opCl(a, b float64) Set { return func(x float64) bool { return a < x && x <= b } } func clOp(a, b float64) Set { return func(x float64) bool { return a <= x && x < b } }
func main() { s := make([]Set, 4) s[0] = Union(opCl(0, 1), clOp(0, 2)) // (0,1] ∪ [0,2) s[1] = Inter(clOp(0, 2), opCl(1, 2)) // [0,2) ∩ (1,2] s[2] = Diff(clOp(0, 3), open(0, 1)) // [0,3) − (0,1) s[3] = Diff(clOp(0, 3), closed(0, 1)) // [0,3) − [0,1]
for i := range s { for x := float64(0); x < 3; x++ { fmt.Printf("%v ∈ s%d: %t\n", x, i, s[i](x)) } fmt.Println() } }</lang> Run in Go Playground.
- Output:
0 ∈ s0: true 1 ∈ s0: true 2 ∈ s0: false 0 ∈ s1: false 1 ∈ s1: false 2 ∈ s1: false 0 ∈ s2: true 1 ∈ s2: true 2 ∈ s2: true 0 ∈ s3: false 1 ∈ s3: false 2 ∈ s3: true
This simple implementation doesn't support lengths so the although the A, B, and A−B sets can be defined and tested (see below), they can't be used to implement the optional part. <lang Go> A := Inter(open(0, 10), func(x float64) bool { return math.Abs(math.Sin(math.Pi*x*x)) > .5 }) B := Inter(open(0, 10), func(x float64) bool { return math.Abs(math.Sin(math.Pi*x)) > .5 }) C := Diff(A, B) // Can't get lengths, can only test for ∈ for x := float64(5.98); x < 6.025; x += 0.01 { fmt.Printf("%.2f ∈ A−B: %t\n", x, C(x)) }</lang>
Haskell
<lang haskell> {- Not so functional representation of R sets (with IEEE Double), in a strange way -}
import Data.List import Data.Maybe
data BracketType = OpenSub | ClosedSub
deriving (Show, Enum, Eq, Ord)
data RealInterval = RealInterval {left :: BracketType, right :: BracketType,
lowerBound :: Double, upperBound :: Double} deriving (Eq)
type RealSet = [RealInterval] posInf = 1.0/0.0 :: Double -- IEEE tricks negInf = (-1.0/0.0) :: Double set_R = RealInterval ClosedSub ClosedSub negInf posInf :: RealInterval
emptySet = [] :: [RealInterval]
instance Show RealInterval where
show x@(RealInterval _ _ y y') | y == y' && (left x == right x) && (left x == ClosedSub) = "{" ++ (show y) ++ "}" | otherwise = [['(', '[']!!(fromEnum $ left x)] ++ (show $ lowerBound x) ++ "," ++ (show $ upperBound x) ++ [[')', ']']!!(fromEnum $ right x)] showList [x] = shows x showList (h:t) = shows h . (" U " ++) . showList t showList [] = (++ "(/)") -- empty set
construct_interval :: Char -> Double -> Double -> Char -> RealInterval construct_interval '(' x y ')' = RealInterval OpenSub OpenSub x y construct_interval '(' x y ']' = RealInterval OpenSub ClosedSub x y construct_interval '[' x y ')' = RealInterval ClosedSub OpenSub x y construct_interval _ x y _ = RealInterval ClosedSub ClosedSub x y
set_is_empty :: RealSet -> Bool set_is_empty rs = (rs == emptySet)
set_in :: Double -> RealSet -> Bool set_in x [] = False set_in x rs =
isJust (find (\s -> ((lowerBound s < x) && (x < upperBound s)) || (x == lowerBound s && left s == ClosedSub) || (x == upperBound s && right s == ClosedSub)) rs)
-- max, min for pairs (double, bracket) max_p :: (Double, BracketType) -> (Double, BracketType) -> (Double, BracketType) min_p :: (Double, BracketType) -> (Double, BracketType) -> (Double, BracketType) max_p p1@(x, y) p2@(x', y')
| x == x' = (x, max y y') -- closed is stronger than open | x < x' = p2 | otherwise = p1
min_p p1@(x, y) p2@(x', y')
| x == x' = (x, min y y') | x < x' = p1 | otherwise = p2
simple_intersection :: RealInterval -> RealInterval -> [RealInterval] simple_intersection ri1@(RealInterval l_ri1 r_ri1 x1 y1) ri2@(RealInterval l_ri2 r_ri2 x2 y2)
| (y1 < x2) || (y2 < x1) = emptySet | (y1 == x2) && ((fromEnum r_ri1) + (fromEnum l_ri2) /= 2) = emptySet | (y2 == x1) && ((fromEnum r_ri2) + (fromEnum l_ri1) /= 2) = emptySet | otherwise = let lb = if x1 == x2 then (x1, min l_ri1 l_ri2) else max_p (x1, l_ri1) (x2, l_ri2) in let rb = min_p (y1, right ri1) (y2, right ri2) in [RealInterval (snd lb) (snd rb) (fst lb) (fst rb)]
simple_union :: RealInterval -> RealInterval -> [RealInterval] simple_union ri1@(RealInterval l_ri1 r_ri1 x1 y1) ri2@(RealInterval l_ri2 r_ri2 x2 y2)
| (y1 < x2) || (y2 < x1) = [ri2, ri1] | (y1 == x2) && ((fromEnum r_ri1) + (fromEnum l_ri2) /= 2) = [ri1, ri2] | (y2 == x1) && ((fromEnum r_ri2) + (fromEnum l_ri1) /= 2) = [ri1, ri2] | otherwise = let lb = if x1 == x2 then (x1, max l_ri1 l_ri2) else min_p (x1, l_ri1) (x2, l_ri2) in let rb = max_p (y1, right ri1) (y2, right ri2) in [RealInterval (snd lb) (snd rb) (fst lb) (fst rb)]
simple_complement :: RealInterval -> [RealInterval] simple_complement ri1@(RealInterval l_ri1 r_ri1 x1 y1) =
[(RealInterval ClosedSub (inv l_ri1) negInf x1), (RealInterval (inv r_ri1) ClosedSub y1 posInf)] where inv OpenSub = ClosedSub inv ClosedSub = OpenSub
set_sort :: RealSet -> RealSet set_sort rs =
sortBy (\s1 s2 -> let (lp, rp) = ((lowerBound s1, left s1), (lowerBound s2, left s2)) in if max_p lp rp == lp then GT else LT) rs
set_simplify :: RealSet -> RealSet set_simplify [] = emptySet set_simplify rs =
concat (map make_empty (set_sort (foldl (\acc ri1 -> (simple_union (head acc) ri1) ++ (tail acc)) [head sorted_rs] sorted_rs))) where sorted_rs = set_sort rs make_empty ri@(RealInterval lb rb x y) | x >= y && (lb /= rb || rb /= ClosedSub) = emptySet | otherwise = [ri]
-- set operations set_complement :: RealSet -> RealSet set_union :: RealSet -> RealSet -> RealSet set_intersection :: RealSet -> RealSet -> RealSet set_difference :: RealSet -> RealSet -> RealSet set_measure :: RealSet -> Double
set_complement rs =
foldl set_intersection [set_R] (map simple_complement rs)
set_union rs1 rs2 =
set_simplify (rs1 ++ rs2)
set_intersection rs1 rs2 =
set_simplify $ concat [simple_intersection s1 s2 | s1 <- rs1, s2 <- rs2]
set_difference rs1 rs2 =
set_intersection (set_complement rs2) rs1
set_measure rs =
foldl (\acc x -> acc + (upperBound x) - (lowerBound x)) 0.0 rs
-- test test = map (\x -> [x]) [construct_interval '(' 0 1 ']', construct_interval '[' 0 2 ')',
construct_interval '[' 0 2 ')', construct_interval '(' 1 2 ']', construct_interval '[' 0 3 ')', construct_interval '(' 0 1 ')', construct_interval '[' 0 3 ')', construct_interval '[' 0 1 ']']
restest = [set_union (test!!0) (test!!1), set_intersection (test!!2) (test!!3),
set_difference (test!!4) (test!!5), set_difference (test!!6) (test!!7)]
isintest s =
mapM_ (\x -> putStrLn ((show x) ++ " is in " ++ (show s) ++ " : " ++ (show (set_in x s)))) [0, 1, 2]
testA = [construct_interval '(' (sqrt (n + (1.0/6))) (sqrt (n + (5.0/6))) ')' | n <- [0..99]] testB = [construct_interval '(' (n + (1.0/6)) (n + (5.0/6)) ')' | n <- [0..9]]
main =
putStrLn ("union " ++ (show (test!!0)) ++ " " ++ (show (test!!1)) ++ " = " ++ (show (restest!!0))) >> putStrLn ("inter " ++ (show (test!!2)) ++ " " ++ (show (test!!3)) ++ " = " ++ (show (restest!!1))) >> putStrLn ("diff " ++ (show (test!!4)) ++ " " ++ (show (test!!5)) ++ " = " ++ (show (restest!!2))) >> putStrLn ("diff " ++ (show (test!!6)) ++ " " ++ (show (test!!7)) ++ " = " ++ (show (restest!!3))) >> mapM_ isintest restest >> putStrLn ("measure: " ++ (show (set_measure (set_difference testA testB))))
</lang>
- Output:
union (0.0,1.0] [0.0,2.0) = [0.0,2.0) inter [0.0,2.0) (1.0,2.0] = (1.0,2.0) diff [0.0,3.0) (0.0,1.0) = {0.0} U [1.0,3.0) diff [0.0,3.0) [0.0,1.0] = (1.0,3.0) 0.0 is in [0.0,2.0) : True 1.0 is in [0.0,2.0) : True 2.0 is in [0.0,2.0) : False 0.0 is in (1.0,2.0) : False 1.0 is in (1.0,2.0) : False 2.0 is in (1.0,2.0) : False 0.0 is in {0.0} U [1.0,3.0) : True 1.0 is in {0.0} U [1.0,3.0) : True 2.0 is in {0.0} U [1.0,3.0) : True 0.0 is in (1.0,3.0) : False 1.0 is in (1.0,3.0) : False 2.0 is in (1.0,3.0) : True measure: 2.0758648411846696
Icon and Unicon
The following only works in Unicon. The code does a few crude simplifications of some representations, but more could be done.
<lang unicon>procedure main(A)
s1 := RealSet("(0,1]").union(RealSet("[0,2)")) s2 := RealSet("[0,2)").intersect(RealSet("(1,2)")) s3 := RealSet("[0,3)").difference(RealSet("(0,1)")) s4 := RealSet("[0,3)").difference(RealSet("[0,1]")) every s := s1|s2|s3|s4 do { every n := 0 to 2 do write(s.toString(),if s.contains(n) then " contains " else " doesn't contain ",n) write() }
end
class Range(a,b,lbnd,rbnd,ltest,rtest)
method contains(x); return ((ltest(a,x),rtest(x,b)),self); end method toString(); return lbnd||a||","||b||rbnd; end method notEmpty(); return (ltest(a,b),rtest(a,b),self); end method makeLTest(); return proc(if lbnd == "(" then "<" else "<=",2); end method makeRTest(); return proc(if rbnd == "(" then "<" else "<=",2); end
method intersect(r) if a < r.a then (na := r.a, nlb := r.lbnd) else if a > r.a then (na := a, nlb := lbnd) else (na := a, nlb := if "(" == (lbnd|r.lbnd) then "(" else "[") if b < r.b then ( nb := b, nrb := rbnd) else if b > r.b then (nb := r.b, nrb := r.rbnd) else (nb := b, nrb := if ")" == (rbnd|r.rbnd) then ")" else "]") range := Range(nlb||na||","||nb||nrb) return range end
method difference(r) if /r then return RealSet(toString()) r1 := lbnd||a||","||min(b,r.a)||map(r.lbnd,"([","])") r2 := map(r.rbnd,")]","[(")||max(a,r.b)||","||b||rbnd return RealSet(r1).union(RealSet(r2)) end
initially(s)
static lbnds, rbnds initial (lbnds := '([', rbnds := '])') if \s then { s ? { lbnd := (tab(upto(lbnds)),move(1)) a := 1(tab(upto(',')),move(1)) b := tab(upto(rbnds)) rbnd := move(1) } ltest := proc(if lbnd == "(" then "<" else "<=",2) rtest := proc(if rbnd == ")" then "<" else "<=",2) }
end
class RealSet(ranges)
method contains(x); return ((!ranges).contains(x), self); end method notEmpty(); return ((!ranges).notEmpty(), self); end
method toString() sep := s := "" every r := (!ranges).toString() do s ||:= .sep || 1(r, sep := " + ") return s end
method clone() newR := RealSet() newR.ranges := (copy(\ranges) | []) return newR end
method union(B) newR := clone() every put(newR.ranges, (!B.ranges).notEmpty()) return newR end
method intersect(B) newR := clone() newR.ranges := [] every (r1 := !ranges, r2 := !B.ranges) do { range := r1.intersect(r2) put(newR.ranges, range.notEmpty()) } return newR end
method difference(B) newR := clone() newR.ranges := [] every (r1 := !ranges, r2 := !B.ranges) do { rs := r1.difference(r2) if rs.notEmpty() then every put(newR.ranges, !rs.ranges) } return newR end
initially(s)
put(ranges := [],Range(\s).notEmpty())
end</lang>
Sample run:
->srn (0,1] + [0,2) contains 0 (0,1] + [0,2) contains 1 (0,1] + [0,2) doesn't contain 2 (1,2) doesn't contain 0 (1,2) doesn't contain 1 (1,2) doesn't contain 2 [0,0] + [1,3) contains 0 [0,0] + [1,3) contains 1 [0,0] + [1,3) contains 2 (1,3) doesn't contain 0 (1,3) doesn't contain 1 (1,3) contains 2 ->
J
In essence, this looks like building a restricted set of statements. So we build a specialized parser and expression builder:
<lang j>has=: 1 :'(interval m)`:6' ing=: `
interval=: 3 :0
if.0<L.y do.y return.end. assert. 5=#words=. ;:y assert. (0 { words) e. ;:'[(' assert. (2 { words) e. ;:',' assert. (4 { words) e. ;:'])' 'lo hi'=.(1 3{0".L:0 words) 'cL cH'=.0 4{words e.;:'[]' (lo&(<`<:@.cL) *. hi&(>`>:@.cH))ing
)
union=: 4 :'(x has +. y has)ing' intersect=: 4 :'(x has *. y has)ing' without=: 4 :'(x has *. [: -. y has)ing'</lang>
With this in place, the required examples look like this:
<lang j> ('(0,1]' union '[0,2)')has 0 1 2 1 1 0
('[0,2)' intersect '(1,2]')has 0 1 2
0 0 0
('[0,3)' without '(0,1]')has 0 1 2
1 0 1
('[0,3)' without '(0,1)')has 0 1 2
1 1 1
('[0,3)' without '[0,1]')has 0 1 2
0 0 1</lang>
Note that without the arguments these wind up being expressions. For example:
<lang j> ('(0,1]' union '[0,2)')has (0&< *. 1&>:) +. 0&<: *. 2&></lang>
In other words, this is a statement built up from inequality terminals (where each inequality is bound to a constant) and the terminals are combined with logical operations.
Optional Work
Empty Set Detection
Here is an alternate formulation which allows detection of empty sets:
<lang j>has=: 1 :'(0 {:: interval m)`:6' ing=: `
edge=: 1&{::&interval edges=: /:~@~.@,&edge contour=: (, 2 (+/%#)\ ])@edge
interval=: 3 :0
if.0<L.y do.y return.end. assert. 5=#words=. ;:y assert. (0 { words) e. ;:'[(' assert. (2 { words) e. ;:',' assert. (4 { words) e. ;:'])' 'lo hi'=.(1 3{0".L:0 words) 'cL cH'=.0 4{words e.;:'[]' (lo&(<`<:@.cL) *. hi&(>`>:@.cH))ing ; lo,hi
)
union=: 4 :'(x has +. y has)ing; x edges y' intersect=: 4 :'(x has *. y has)ing; x edges y' without=: 4 :'(x has *. [: -. y has)ing; x edges y' in=: 4 :'y has x' isEmpty=: 1 -.@e. contour in ]</lang>
The above examples work identically with this version, but also:
<lang j> isEmpty '(0,1]' union '[0,2)' 0
isEmpty '[0,2)' intersect '(1,2]'
0
isEmpty '[0,2)' intersect '(2,3]'
1
isEmpty '[0,2)' intersect '[2,3]'
1
isEmpty '[0,2]' intersect '[2,3]'
0</lang>
Note that the the set operations no longer return a simple verb -- instead, they return a pair, where the first element represents the verb and the second element is a list of interval boundaries. We can tell if two adjacent bounds, from this list, bound a valid interval by checking any point between them.
Length of Set Difference
The optional work centers around expressions where the absolute value of sin pi * n is 0.5. It would be nice if J had an arcsine which gave all values within a range, but it does not have that. So:
<lang j> 1p_1 * _1 o. 0.5 0.166667</lang>
(Note on notation: 1 o. is sine in J, and 2 o. is cosine -- the mnemonic is that sine is an odd function and cosine is an even function, the practical value is that sine, cosine and sine/cosine pairs can all be generated from the same "real" valued function. Similarly, _1 o. is arcsine and _2 o. is arcsine. Also 1p_1 is the reciprocal of pi. So the above tells us that the principal value for arc sine 0.5 is one sixth.)
<lang j> (#~ 0.5 = 1 |@o. 1r6p1&*) i. 30 1 5 7 11 13 17 19 23 25 29
2 -~/\ (#~ 0.5 = 1 |@o. 1r6p1&*) i. 30
4 2 4 2 4 2 4 2 4</lang>
Here we see the integers which when multiplied by pi/6 give 0.5 for the absolute value of the sine, and their first difference. Thus:
<lang j>zeros0toN=: ((>: # ])[:+/\1,$&4 2@<.)&.(6&*)</lang>
is a function to generate the values which correspond to the boundaries of the intervals we want:
<lang j>zB=: zeros0toN 10 zA=: zeros0toN&.*: 10
zA
0.408248 0.912871 1.08012 1.35401 1.47196 1.68325 1.77951 1.95789 2.04124 2.1984...
zB
0.166667 0.833333 1.16667 1.83333 2.16667 2.83333 3.16667 3.83333 4.16667 4.8333...
#zA
200
#zB
20</lang>
And, here are the edges of the sets of intervals we need to consider.
To find the length of the the set A-B we can find the length of set A and subtract the length of the set A-B:
<lang j> (+/_2 -~/\zA) - +/,0>.zA (<.&{: - >.&{.)"1/&(_2 ]\ ]) zB 2.07586</lang>
Here, we have paired adjacent elements from the zero bounding list (non-overlapping infixes of length 2). For set A's length we sum the results of subtracting the smaller number of the pair from the larger. For set A-B's length we consider each combination of pairs from A and B and subtract the larger of the beginning values from the smaller of the ending values (and ignore any negative results).
Alternatively, if we use the set implementation with empty set detection, and the following definitions:
<lang j>intervalSet=: interval@('[',[,',',],')'"_)&": A=: union/_2 intervalSet/\ zA B=: union/_2 intervalSet/\ zB diff=: A without B</lang>
We can replace the above sentence to compute the length of the difference with:
<lang j> +/ ((2 (+/%#)\ edge diff) in diff) * 2 -~/\ edge diff 2.07588</lang>
(Note that this result is not exactly the same as the previous result. Determining why would be an interesting exercise in numerical analysis.)
Java
<lang java>import java.util.Objects; import java.util.function.Predicate;
public class RealNumberSet {
public enum RangeType { CLOSED, BOTH_OPEN, LEFT_OPEN, RIGHT_OPEN, }
public static class RealSet { private Double low; private Double high; private Predicate<Double> predicate; private double interval = 0.00001;
public RealSet(Double low, Double high, Predicate<Double> predicate) { this.low = low; this.high = high; this.predicate = predicate; }
public RealSet(Double start, Double end, RangeType rangeType) { this(start, end, d -> { switch (rangeType) { case CLOSED: return start <= d && d <= end; case BOTH_OPEN: return start < d && d < end; case LEFT_OPEN: return start < d && d <= end; case RIGHT_OPEN: return start <= d && d < end; default: throw new IllegalStateException("Unhandled range type encountered."); } }); }
public boolean contains(Double d) { return predicate.test(d); }
public RealSet union(RealSet other) { double low2 = Math.min(low, other.low); double high2 = Math.max(high, other.high); return new RealSet(low2, high2, d -> predicate.or(other.predicate).test(d)); }
public RealSet intersect(RealSet other) { double low2 = Math.min(low, other.low); double high2 = Math.max(high, other.high); return new RealSet(low2, high2, d -> predicate.and(other.predicate).test(d)); }
public RealSet subtract(RealSet other) { return new RealSet(low, high, d -> predicate.and(other.predicate.negate()).test(d)); }
public double length() { if (low.isInfinite() || high.isInfinite()) return -1.0; // error value if (high <= low) return 0.0; Double p = low; int count = 0; do { if (predicate.test(p)) count++; p += interval; } while (p < high); return count * interval; }
public boolean isEmpty() { if (Objects.equals(high, low)) { return predicate.negate().test(low); } return length() == 0.0; } }
public static void main(String[] args) { RealSet a = new RealSet(0.0, 1.0, RangeType.LEFT_OPEN); RealSet b = new RealSet(0.0, 2.0, RangeType.RIGHT_OPEN); RealSet c = new RealSet(1.0, 2.0, RangeType.LEFT_OPEN); RealSet d = new RealSet(0.0, 3.0, RangeType.RIGHT_OPEN); RealSet e = new RealSet(0.0, 1.0, RangeType.BOTH_OPEN); RealSet f = new RealSet(0.0, 1.0, RangeType.CLOSED); RealSet g = new RealSet(0.0, 0.0, RangeType.CLOSED);
for (int i = 0; i <= 2; i++) { Double dd = (double) i; System.out.printf("(0, 1] ∪ [0, 2) contains %d is %s\n", i, a.union(b).contains(dd)); System.out.printf("[0, 2) ∩ (1, 2] contains %d is %s\n", i, b.intersect(c).contains(dd)); System.out.printf("[0, 3) − (0, 1) contains %d is %s\n", i, d.subtract(e).contains(dd)); System.out.printf("[0, 3) − [0, 1] contains %d is %s\n", i, d.subtract(f).contains(dd)); System.out.println(); }
System.out.printf("[0, 0] is empty is %s\n", g.isEmpty()); System.out.println();
RealSet aa = new RealSet( 0.0, 10.0, x -> (0.0 < x && x < 10.0) && Math.abs(Math.sin(Math.PI * x * x)) > 0.5 ); RealSet bb = new RealSet( 0.0, 10.0, x -> (0.0 < x && x < 10.0) && Math.abs(Math.sin(Math.PI * x)) > 0.5 ); RealSet cc = aa.subtract(bb); System.out.printf("Approx length of A - B is %f\n", cc.length()); }
}</lang>
- Output:
(0, 1] ∪ [0, 2) contains 0 is true [0, 2) ∩ (1, 2] contains 0 is false [0, 3) − (0, 1) contains 0 is true [0, 3) − [0, 1] contains 0 is false (0, 1] ∪ [0, 2) contains 1 is true [0, 2) ∩ (1, 2] contains 1 is false [0, 3) − (0, 1) contains 1 is true [0, 3) − [0, 1] contains 1 is false (0, 1] ∪ [0, 2) contains 2 is false [0, 2) ∩ (1, 2] contains 2 is false [0, 3) − (0, 1) contains 2 is true [0, 3) − [0, 1] contains 2 is true [0, 0] is empty is false Approx length of A - B is 2.075870
JavaScript
<lang javascript> function realSet(set1, set2, op, values) {
const makeSet=(set0)=>{ let res = [] if(set0.rangeType===0){ for(let i=set0.low;i<=set0.high;i++) res.push(i); } else if (set0.rangeType===1) { for(let i=set0.low+1;i<set0.high;i++) res.push(i); } else if(set0.rangeType===2){ for(let i=set0.low+1;i<=set0.high;i++) res.push(i); } else { for(let i=set0.low;i<set0.high;i++) res.push(i); } return res; } let res = [],finalSet=[]; set1 = makeSet(set1); set2 = makeSet(set2); if(op==="union") finalSet = [...new Set([...set1,...set2])]; else if(op==="intersect") { for(let i=0;i<set1.length;i++) if(set1.indexOf(set2[i])!==-1) finalSet.push(set2[i]); } else { for(let i=0;i<set2.length;i++) if(set1.indexOf(set2[i])===-1) finalSet.push(set2[i]);
for(let i=0;i<set1.length;i++) if(set2.indexOf(set1[i])===-1) finalSet.push(set1[i]); } for(let i=0;i<values.length;i++){ if(finalSet.indexOf(values[i])!==-1) res.push(true); else res.push(false); } return res;
} </lang>
Julia
<lang Julia> """
struct ConvexRealSet
Convex real set (similar to a line segment). Parameters: lower bound, upper bound: floating point numbers
includelower, includeupper: boolean true or false to indicate whether the set has a closed boundary (set to true) or open (set to false).
""" mutable struct ConvexRealSet
lower::Float64 includelower::Bool upper::Float64 includeupper::Bool function ConvexRealSet(lo, up, incllo, inclup) this = new() this.upper = Float64(up) this.lower = Float64(lo) this.includelower = incllo this.includeupper = inclup this end
end
function ∈(s, xelem)
x = Float64(xelem) if(x == s.lower) if(s.includelower) return true else return false end elseif(x == s.upper) if(s.includeupper) return true else return false end end s.lower < x && x < s.upper
end
⋃(aset, bset, x) = (∈(aset, x) || ∈(bset, x))
⋂(aset, bset, x) = (∈(aset, x) && ∈(bset, x))
-(aset, bset, x) = (∈(aset, x) && !∈(bset, x))
isempty(s::ConvexRealSet) = (s.lower > s.upper) ||
((s.lower == s.upper) && !s.includeupper && !s.includelower)
const s1 = ConvexRealSet(0.0, 1.0, false, true)
const s2 = ConvexRealSet(0.0, 2.0, true, false)
const s3 = ConvexRealSet(1.0, 2.0, false, true)
const s4 = ConvexRealSet(0.0, 3.0, true, false)
const s5 = ConvexRealSet(0.0, 1.0, false, false)
const s6 = ConvexRealSet(0.0, 1.0, true, true)
const sempty = ConvexRealSet(0.0, -1.0, true, true)
const testlist = [0, 1, 2]
function testconvexrealset()
for i in testlist println("Testing with x = $i.\nResults:") println(" (0, 1] ∪ [0, 2): $(⋃(s1, s2, i))") println(" [0, 2) ∩ (1, 2]: $(⋂(s2, s3, i))") println(" [0, 3) − (0, 1): $(-(s4, s5, i))") println(" [0, 3) − [0, 1]: $(-(s4, s6, i))\n") end print("The set sempty is ") println(isempty(sempty) ? "empty." : "not empty.")
end
testconvexrealset()
</lang>
- Output:
Testing with x = 0. Results:
(0, 1] ∪ [0, 2): true [0, 2) ∩ (1, 2]: false [0, 3) − (0, 1): true [0, 3) − [0, 1]: falseTesting with x = 1. Results:
(0, 1] ∪ [0, 2): true [0, 2) ∩ (1, 2]: false [0, 3) − (0, 1): true [0, 3) − [0, 1]: falseTesting with x = 2. Results:
(0, 1] ∪ [0, 2): false [0, 2) ∩ (1, 2]: false [0, 3) − (0, 1): true [0, 3) − [0, 1]: trueThe set sempty is empty.
Kotlin
The RealSet class has two constructors - a primary one which creates an object for an arbitrary predicate and a secondary one which creates an object for a simple range by generating the appropriate predicate and then invoking the primary one.
As far as the optional work is concerned, I decided to add a length property which gives only an approximate result. Basically, it works by keeping track of the low and high values of the set and then counting points at successive small intervals between these limits which satisfy the predicate. An isEmpty() function has also been added but as this depends, to some extent, on the length property it is not 100% reliable.
Clearly, the above approach is only suitable for sets with narrow ranges (as we have here) but does have the merit of not over-complicating the basic class. <lang scala>// version 1.1.4-3
typealias RealPredicate = (Double) -> Boolean
enum class RangeType { CLOSED, BOTH_OPEN, LEFT_OPEN, RIGHT_OPEN }
class RealSet(val low: Double, val high: Double, val predicate: RealPredicate) {
constructor (start: Double, end: Double, rangeType: RangeType): this(start, end, when (rangeType) { RangeType.CLOSED -> fun(d: Double) = d in start..end RangeType.BOTH_OPEN -> fun(d: Double) = start < d && d < end RangeType.LEFT_OPEN -> fun(d: Double) = start < d && d <= end RangeType.RIGHT_OPEN -> fun(d: Double) = start <= d && d < end } )
fun contains(d: Double) = predicate(d)
infix fun union(other: RealSet): RealSet { val low2 = minOf(low, other.low) val high2 = maxOf(high, other.high) return RealSet(low2, high2) { predicate(it) || other.predicate(it) } } infix fun intersect(other: RealSet): RealSet { val low2 = maxOf(low, other.low) val high2 = minOf(high, other.high) return RealSet(low2, high2) { predicate(it) && other.predicate(it) } }
infix fun subtract(other: RealSet) = RealSet(low, high) { predicate(it) && !other.predicate(it) }
var interval = 0.00001
val length: Double get() { if (!low.isFinite() || !high.isFinite()) return -1.0 // error value if (high <= low) return 0.0 var p = low var count = 0 do { if (predicate(p)) count++ p += interval } while (p < high) return count * interval }
fun isEmpty() = if (high == low) !predicate(low) else length == 0.0
}
fun main(args: Array<String>) {
val a = RealSet(0.0, 1.0, RangeType.LEFT_OPEN) val b = RealSet(0.0, 2.0, RangeType.RIGHT_OPEN) val c = RealSet(1.0, 2.0, RangeType.LEFT_OPEN) val d = RealSet(0.0, 3.0, RangeType.RIGHT_OPEN) val e = RealSet(0.0, 1.0, RangeType.BOTH_OPEN) val f = RealSet(0.0, 1.0, RangeType.CLOSED) val g = RealSet(0.0, 0.0, RangeType.CLOSED)
for (i in 0..2) { val dd = i.toDouble() println("(0, 1] ∪ [0, 2) contains $i is ${(a union b).contains(dd)}") println("[0, 2) ∩ (1, 2] contains $i is ${(b intersect c).contains(dd)}") println("[0, 3) − (0, 1) contains $i is ${(d subtract e).contains(dd)}") println("[0, 3) − [0, 1] contains $i is ${(d subtract f).contains(dd)}\n") }
println("[0, 0] is empty is ${g.isEmpty()}\n")
val aa = RealSet(0.0, 10.0) { x -> (0.0 < x && x < 10.0) && Math.abs(Math.sin(Math.PI * x * x)) > 0.5 } val bb = RealSet(0.0, 10.0) { x -> (0.0 < x && x < 10.0) && Math.abs(Math.sin(Math.PI * x)) > 0.5 } val cc = aa subtract bb println("Approx length of A - B is ${cc.length}")
}</lang>
- Output:
(0, 1] ∪ [0, 2) contains 0 is true [0, 2) ∩ (1, 2] contains 0 is false [0, 3) − (0, 1) contains 0 is true [0, 3) − [0, 1] contains 0 is false (0, 1] ∪ [0, 2) contains 1 is true [0, 2) ∩ (1, 2] contains 1 is false [0, 3) − (0, 1) contains 1 is true [0, 3) − [0, 1] contains 1 is false (0, 1] ∪ [0, 2) contains 2 is false [0, 2) ∩ (1, 2] contains 2 is false [0, 3) − (0, 1) contains 2 is true [0, 3) − [0, 1] contains 2 is true [0, 0] is empty is false Approx length of A - B is 2.07587
Lua
<lang lua>function createSet(low,high,rt)
local l,h = tonumber(low), tonumber(high) if l and h then local t = {low=l, high=h}
if type(rt) == "string" then if rt == "open" then t.contains = function(d) return low< d and d< high end elseif rt == "closed" then t.contains = function(d) return low<=d and d<=high end elseif rt == "left" then t.contains = function(d) return low< d and d<=high end elseif rt == "right" then t.contains = function(d) return low<=d and d< high end else error("Unknown range type: "..rt) end elseif type(rt) == "function" then t.contains = rt else error("Unable to find a range type or predicate") end
t.union = function(o) local l2 = math.min(l, o.low) local h2 = math.min(h, o.high) local p = function(d) return t.contains(d) or o.contains(d) end return createSet(l2, h2, p) end
t.intersect = function(o) local l2 = math.min(l, o.low) local h2 = math.min(h, o.high) local p = function(d) return t.contains(d) and o.contains(d) end return createSet(l2, h2, p) end
t.subtract = function(o) local l2 = math.min(l, o.low) local h2 = math.min(h, o.high) local p = function(d) return t.contains(d) and not o.contains(d) end return createSet(l2, h2, p) end
t.length = function() if h <= l then return 0.0 end local p = l local count = 0 local interval = 0.00001 repeat if t.contains(p) then count = count + 1 end p = p + interval until p>=high return count * interval end
t.empty = function() if l == h then return not t.contains(low) end return t.length() == 0.0 end
return t else error("Either '"..low.."' or '"..high.."' is not a number") end
end
local a = createSet(0.0, 1.0, "left") local b = createSet(0.0, 2.0, "right") local c = createSet(1.0, 2.0, "left") local d = createSet(0.0, 3.0, "right") local e = createSet(0.0, 1.0, "open") local f = createSet(0.0, 1.0, "closed") local g = createSet(0.0, 0.0, "closed")
for i=0,2 do
print("(0, 1] union [0, 2) contains "..i.." is "..tostring(a.union(b).contains(i))) print("[0, 2) intersect (1, 2] contains "..i.." is "..tostring(b.intersect(c).contains(i))) print("[0, 3) - (0, 1) contains "..i.." is "..tostring(d.subtract(e).contains(i))) print("[0, 3) - [0, 1] contains "..i.." is "..tostring(d.subtract(f).contains(i))) print()
end
print("[0, 0] is empty is "..tostring(g.empty())) print()
local aa = createSet(
0.0, 10.0, function(x) return (0.0<x and x<10.0) and math.abs(math.sin(math.pi * x * x)) > 0.5 end
) local bb = createSet(
0.0, 10.0, function(x) return (0.0<x and x<10.0) and math.abs(math.sin(math.pi * x)) > 0.5 end
) local cc = aa.subtract(bb) print("Approx length of A - B is "..cc.length())</lang>
- Output:
(0, 1] union [0, 2) contains 0 is true [0, 2) intersect (1, 2] contains 0 is false [0, 3) - (0, 1) contains 0 is true [0, 3) - [0, 1] contains 0 is false (0, 1] union [0, 2) contains 1 is true [0, 2) intersect (1, 2] contains 1 is false [0, 3) - (0, 1) contains 1 is true [0, 3) - [0, 1] contains 1 is false (0, 1] union [0, 2) contains 2 is false [0, 2) intersect (1, 2] contains 2 is false [0, 3) - (0, 1) contains 2 is true [0, 3) - [0, 1] contains 2 is true [0, 0] is empty is false Approx length of A - B is 2.07587
Mathematica
<lang Mathematica>(* defining functions *) setcc[a_, b_] := a <= x <= b setoo[a_, b_] := a < x < b setco[a_, b_] := a <= x < b setoc[a_, b_] := a < x <= b setSubtract[s1_, s2_] := s1 && Not[s2]; (* new function; subtraction not built in *) inSetQ[y_, set_] := set /. x -> y (* testing sets *) set1 = setoc[0, 1] || setco[0, 2] (* union built in as || shortcut (OR) *); Print[set1] Print["First trial set, (0, 1] ∪ [0, 2) , testing for {0,1,2}:"] Print[inSetQ[#, set1] & /@ {0, 1, 2}] set2 = setco[0, 2] && setoc[1, 2]; (* intersection built in as && shortcut (AND) *) Print[] Print[set2] Print["Second trial set, [0, 2) ∩ (1, 2], testing for {0,1,2}:"] Print[inSetQ[#, set2] & /@ {0, 1, 2}] Print[] set3 = setSubtract[setco[0, 3], setoo[0, 1]]; Print[set3] Print["Third trial set, [0, 3) \[Minus] (0, 1), testing for {0,1,2}"] Print[inSetQ[#, set3] & /@ {0, 1, 2}] Print[] set4 = setSubtract[setco[0, 3], setcc[0, 1]]; Print[set4] Print["Fourth trial set, [0,3)\[Minus][0,1], testing for {0,1,2}:"] Print[inSetQ[#, set4] & /@ {0, 1, 2}]</lang>
- Output:
0<x<=1||0<=x<2 First trial set, (0, 1] ∪ [0, 2) , testing for {0,1,2}: {True,True,False} 0<=x<2&&1<x<=2 Second trial set, [0, 2) ∩ (1, 2], testing for {0,1,2}: {False,False,False} 0<=x<3&&!0<x<1 Third trial set, [0, 3) \[Minus] (0, 1), testing for {0,1,2} {True,True,True} 0<=x<3&&!0<=x<=1 Fourth trial set, [0,3)\[Minus][0,1], testing for {0,1,2}: {False,False,True}
Nim
<lang Nim>import math, strformat, sugar
type
RealPredicate = (float) -> bool
RangeType {.pure} = enum Closed, BothOpen, LeftOpen, RightOpen
RealSet = object low, high: float predicate: RealPredicate
proc initRealSet(slice: Slice[float]; rangeType: RangeType): RealSet =
result = RealSet(low: slice.a, high: slice.b) result.predicate = case rangeType of Closed: (x: float) => x in slice of BothOpen: (x: float) => slice.a < x and x < slice.b of LeftOpen: (x: float) => slice.a < x and x <= slice.b of RightOpen: (x: float) => slice.a <= x and x < slice.b
proc contains(s: RealSet; val: float): bool =
## Defining "contains" makes operator "in" available. s.predicate(val)
proc `+`(s1, s2: RealSet): RealSet =
RealSet(low: min(s1.low, s2.low), high: max(s1.high, s2.high), predicate: (x:float) => s1.predicate(x) or s2.predicate(x))
proc `*`(s1, s2: RealSet): RealSet =
RealSet(low: max(s1.low, s2.low), high: min(s1.high, s2.high), predicate: (x:float) => s1.predicate(x) and s2.predicate(x))
proc `-`(s1, s2: RealSet): RealSet =
RealSet(low: s1.low, high: s1.high, predicate: (x:float) => s1.predicate(x) and not s2.predicate(x))
const Interval = 0.00001
proc length(s: RealSet): float =
if s.low.classify() in {fcInf, fcNegInf} or s.high.classify() in {fcInf, fcNegInf}: return Inf if s.high <= s.low: return 0 var p = s.low var count = 0.0 while p < s.high: if s.predicate(p): count += 1 p += Interval result = count * Interval
proc isEmpty(s: RealSet): bool =
if s.high == s.low: not s.predicate(s.low) else: s.length == 0
when isMainModule:
let a = initRealSet(0.0..1.0, LeftOpen) b = initRealSet(0.0..2.0, RightOpen) c = initRealSet(1.0..2.0, LeftOpen) d = initRealSet(0.0..3.0, RightOpen) e = initRealSet(0.0..1.0, BothOpen) f = initRealSet(0.0..1.0, Closed) g = initRealSet(0.0..0.0, Closed)
for n in 0..2: let x = n.toFloat echo &"{n} ∊ (0, 1] ∪ [0, 2) is {x in (a + b)}" echo &"{n} ∊ [0, 2) ∩ (1, 2] is {x in (b * c)}" echo &"{n} ∊ [0, 3) − (0, 1) is {x in (d - e)}" echo &"{n} ∊ [0, 3) − [0, 1] is {x in (d - f)}\n"
echo &"[0, 0] is empty is {g.isEmpty()}.\n"
let aa = RealSet(low: 0, high: 10, predicate: (x: float) => 0 < x and x < 10 and abs(sin(PI * x * x)) > 0.5) bb = RealSet(low: 0, high: 10, predicate: (x: float) => 0 < x and x < 10 and abs(sin(PI * x)) > 0.5) cc = aa - bb
echo &"Approximative length of A - B is {cc.length}."</lang>
- Output:
0 ∊ (0, 1] ∪ [0, 2) is true 0 ∊ [0, 2) ∩ (1, 2] is false 0 ∊ [0, 3) − (0, 1) is true 0 ∊ [0, 3) − [0, 1] is false 1 ∊ (0, 1] ∪ [0, 2) is true 1 ∊ [0, 2) ∩ (1, 2] is false 1 ∊ [0, 3) − (0, 1) is true 1 ∊ [0, 3) − [0, 1] is false 2 ∊ (0, 1] ∪ [0, 2) is false 2 ∊ [0, 2) ∩ (1, 2] is false 2 ∊ [0, 3) − (0, 1) is true 2 ∊ [0, 3) − [0, 1] is true [0, 0] is empty is false. Approximative length of A - B is 2.07587.
PARI/GP
Define some sets and use built-in functions: <lang parigp>set11(x,a,b)=select(x -> a <= x && x <= b, x); set01(x,a,b)=select(x -> a < x && x <= b, x); set10(x,a,b)=select(x -> a <= x && x < b, x); set00(x,a,b)=select(x -> a < x && x < b, x);
V = [0, 1, 2];
setunion(set01(V, 0, 1), set10(V, 0, 2))
setintersect(set10(V, 0, 2), set01(V, 1, 2))
setminus(set10(V, 0, 3), set00(V, 0, 1)) setminus(set10(V, 0, 3), set11(V, 0, 1))</lang>
Output:
[0, 1] [] [0, 1, 2] [2]
Perl
<lang perl>use utf8;
- numbers used as boundaries to real sets. Each has 3 components:
- the real value x;
- a +/-1 indicating if it's x + ϵ or x - ϵ
- a 0/1 indicating if it's the left border or right border
- e.g. "[1.5, ..." is written "1.5, -1, 0", while "..., 2)" is "2, -1, 1"
package BNum;
use overload ( '""' => \&_str, '<=>' => \&_cmp, );
sub new { my $self = shift; bless [@_], ref $self || $self }
sub flip { my @a = @{+shift}; $a[2] = !$a[2]; bless \@a }
my $brackets = qw/ [ ( ) ] /; sub _str { my $v = sprintf "%.2f", $_[0][0]; $_[0][2] ? $v . ($_[0][1] == 1 ? "]" : ")") : ($_[0][1] == 1 ? "(" : "[" ) . $v; }
sub _cmp { my ($a, $b, $swap) = @_;
# if one of the argument is a normal number if ($swap) { return -_ncmp($a, $b) } if (!ref $b || !$b->isa(__PACKAGE__)) { return _ncmp($a, $b) }
$a->[0] <=> $b->[0] || $a->[1] <=> $b->[1] }
sub _ncmp { # $a is a BNum, $b is something comparable to a real my ($a, $b) = @_; $a->[0] <=> $b || $a->[1] <=> 0 }
package RealSet; use Carp; use overload ( '""' => \&_str, '|' => \&_or, '&' => \&_and, '~' => \&_neg, '-' => \&_diff, 'bool' => \¬_empty, # set is true if not empty );
my %pm = qw/ [ -1 ( 1 ) -1 ] 1 /; sub range { my ($cls, $a, $b, $spec) = @_; $spec =~ /^( \[ | \( )( \) | \] )$/x or croak "bad spec $spec";
$a = BNum->new($a, $pm{$1}, 0); $b = BNum->new($b, $pm{$2}, 1); normalize($a < $b ? [$a, $b] : []) }
sub normalize { my @a = @{+shift}; # remove invalid or duplicate borders, such as "[2, 1]" or "3) [3" # note that "(a" == "a]" and "a)" == "[a", but "a)" < "(a" and # "[a" < "a]" for (my $i = $#a; $i > 0; $i --) { splice @a, $i - 1, 2 if $a[$i] <= $a[$i - 1] } bless \@a }
sub not_empty { scalar @{ normalize shift } }
sub _str { my (@a, @s) = @{+shift} or return '()'; join " ∪ ", map { shift(@a).", ".shift(@a) } 0 .. $#a/2 }
sub _or { # we may have nested ranges now; let only outmost ones survive my $d = 0; normalize [ map { $_->[2] ? --$d ? () : ($_) : $d++ ? () : ($_) } sort{ $a <=> $b } @{+shift}, @{+shift} ]; }
sub _neg { normalize [ BNum->new('-inf', 1, 0), map($_->flip, @{+shift}), BNum->new('inf', -1, 1), ] }
sub _and { my $d = 0; normalize [ map { $_->[2] ? --$d ? ($_) : () : $d++ ? ($_) : () } sort{ $a <=> $b } @{+shift}, @{+shift} ]; }
sub _diff { shift() & ~shift() }
sub has { my ($a, $b) = @_; for (my $i = 0; $i < $#$a; $i += 2) { return 1 if $a->[$i] <= $b && $b <= $a->[$i + 1] } return 0 }
sub len { my ($a, $l) = shift; for (my $i = 0; $i < $#$a; $i += 2) { $l += $a->[$i+1][0] - $a->[$i][0] } return $l }
package main; use List::Util 'reduce';
sub rng { RealSet->range(@_) } my @sets = ( rng(0, 1, '(]') | rng(0, 2, '[)'), rng(0, 2, '[)') & rng(0, 2, '(]'), rng(0, 3, '[)') - rng(0, 1, '()'), rng(0, 3, '[)') - rng(0, 1, '[]'), );
for my $i (0 .. $#sets) { print "Set $i = ", $sets[$i], ": "; for (0 .. 2) { print "has $_; " if $sets[$i]->has($_); } print "\n"; }
- optional task
print "\n####\n"; sub brev { # show only head and tail if string too long my $x = shift; return $x if length $x < 60; substr($x, 0, 30)." ... ".substr($x, -30, 30) }
- "|sin(x)| > 1/2" means (n + 1/6) pi < x < (n + 5/6) pi
my $x = reduce { $a | $b } map(rng(sqrt($_ + 1./6), sqrt($_ + 5./6), '()'), 0 .. 101); $x &= rng(0, 10, '()');
print "A\t", '= {x | 0 < x < 10 and |sin(π x²)| > 1/2 }', "\n\t= ", brev($x), "\n";
my $y = reduce { $a | $b } map { rng($_ + 1./6, $_ + 5./6, '()') } 0 .. 11; $y &= rng(0, 10, '()');
print "B\t", '= {x | 0 < x < 10 and |sin(π x)| > 1/2 }', "\n\t= ", brev($y), "\n";
my $z = $x - $y; print "A - B\t= ", brev($z), "\n\tlength = ", $z->len, "\n"; print $z ? "not empty\n" : "empty\n";</lang>output<lang>Set 0 = [0.00, 2.00): has 0; has 1; Set 1 = (0.00, 2.00): has 1; Set 2 = [0.00, 0.00] ∪ [1.00, 3.00): has 0; has 1; has 2; Set 3 = (1.00, 3.00): has 2;
A = {x | 0 < x < 10 and |sin(π x²)| > 1/2 }
= (0.41, 0.91) ∪ (1.08, 1.35) ∪ ... ∪ (9.91, 9.94) ∪ (9.96, 9.99)
B = {x | 0 < x < 10 and |sin(π x)| > 1/2 }
= (0.17, 0.83) ∪ (1.17, 1.83) ∪ ... ∪ (8.17, 8.83) ∪ (9.17, 9.83)
A - B = [0.83, 0.91) ∪ (1.08, 1.17] ∪ ... ∪ (9.91, 9.94) ∪ (9.96, 9.99)
length = 2.07586484118467 not empty</lang>
Phix
<lang Phix>enum ID,ARGS function cf(sequence f, atom x) return call_func(f[ID],f[ARGS]&x) end function function Union(sequence a, b, atom x) return cf(a,x) or cf(b,x) end function function Inter(sequence a, b, atom x) return cf(a,x) and cf(b,x) end function function Diffr(sequence a, b, atom x) return cf(a,x) and not cf(b,x) end function function OpOp(atom a, b, x) return a < x and x < b end function function ClCl(atom a, b, x) return a <= x and x <= b end function function OpCl(atom a, b, x) return a < x and x <= b end function function ClOp(atom a, b, x) return a <= x and x < b end function
-- expected -- ---- desc ----, 0 1 2, --------------- set method --------------- constant s = {{"(0,1] u [0,2)", {1,1,0}, {Union,{{OpCl,{0,1}},{ClOp,{0,2}}}}},
{"[0,2) n (1,2]", {0,0,0}, {Inter,{{ClOp,{0,2}},{OpCl,{1,2}}}}}, {"[0,3) - (0,1)", {1,1,1}, {Diffr,{{ClOp,{0,3}},{OpOp,{0,1}}}}}, {"[0,3) - [0,1]", {0,0,1}, {Diffr,{{ClOp,{0,3}},{ClCl,{0,1}}}}}}
for i=1 to length(s) do
sequence {desc, expect, method} = s[i] for x=0 to 2 do bool r = cf(method,x) string error = iff(r!=expect[x+1]?"error":"") printf(1,"%d in %s : %t %s\n", {x, desc, r, error}) end for printf(1,"\n")
end for</lang>
- Output:
0 in (0,1] u [0,2) : true 1 in (0,1] u [0,2) : true 2 in (0,1] u [0,2) : false 0 in [0,2) n (1,2] : false 1 in [0,2) n (1,2] : false 2 in [0,2) n (1,2] : false 0 in [0,3) - (0,1) : true 1 in [0,3) - (0,1) : true 2 in [0,3) - (0,1) : true 0 in [0,3) - [0,1] : false 1 in [0,3) - [0,1] : false 2 in [0,3) - [0,1] : true
Extra credit - also translated from Go, but with an extended loop and crude summation, inspired by Java/Kotlin. <lang Phix>function aspxx(atom x) return abs(sin(PI*x*x))>0.5 end function function aspx(atom x) return abs(sin(PI*x)) >0.5 end function
constant A = {Inter,{{OpOp,{0,10}},{aspxx,{}}}},
B = {Inter,{{OpOp,{0,10}},{aspx,{}}}}, C = {Diffr,{A,B}}
atom x = 0, step = 0.00001, count = 0 while x<=10 do
count += cf(C,x) x += step
end while printf(1,"Approximate length of A-B: %.5f\n",{count*step})</lang>
- Output:
Approximate length of A-B: 2.07587
Python
<lang python>class Setr():
def __init__(self, lo, hi, includelo=True, includehi=False): self.eqn = "(%i<%sX<%s%i)" % (lo, '=' if includelo else , '=' if includehi else , hi)
def __contains__(self, X): return eval(self.eqn, locals())
# union def __or__(self, b): ans = Setr(0,0) ans.eqn = "(%sor%s)" % (self.eqn, b.eqn) return ans
# intersection def __and__(self, b): ans = Setr(0,0) ans.eqn = "(%sand%s)" % (self.eqn, b.eqn) return ans
# difference def __sub__(self, b): ans = Setr(0,0) ans.eqn = "(%sand not%s)" % (self.eqn, b.eqn) return ans
def __repr__(self): return "Setr%s" % self.eqn
sets = [
Setr(0,1, 0,1) | Setr(0,2, 1,0), Setr(0,2, 1,0) & Setr(1,2, 0,1), Setr(0,3, 1,0) - Setr(0,1, 0,0), Setr(0,3, 1,0) - Setr(0,1, 1,1),
] settexts = '(0, 1] ∪ [0, 2);[0, 2) ∩ (1, 2];[0, 3) − (0, 1);[0, 3) − [0, 1]'.split(';')
for s,t in zip(sets, settexts):
print("Set %s %s. %s" % (t, ', '.join("%scludes %i" % ('in' if v in s else 'ex', v) for v in range(3)), s.eqn))</lang>
- Output
Set (0, 1] ∪ [0, 2) includes 0, includes 1, excludes 2. ((0<X<=1)or(0<=X<2)) Set [0, 2) ∩ (1, 2] excludes 0, excludes 1, excludes 2. ((0<=X<2)and(1<X<=2)) Set [0, 3) − (0, 1) includes 0, includes 1, includes 2. ((0<=X<3)and not(0<X<1)) Set [0, 3) − [0, 1] excludes 0, excludes 1, includes 2. ((0<=X<3)and not(0<=X<=1))
Racket
This is a simple representation of sets as functions (so obviously no good way to the the extra set length). <lang Racket>
- lang racket
- Use a macro to allow infix operators
(require (only-in racket [#%app #%%app])) (define-for-syntax infixes '()) (define-syntax (definfix stx)
(syntax-case stx () [(_ (x . xs) body ...) #'(definfix x (λ xs body ...))] [(_ x body) (begin (set! infixes (cons #'x infixes)) #'(define x body))]))
(define-syntax (#%app stx)
(syntax-case stx () [(_ X op Y) (and (identifier? #'op) (ormap (λ(o) (free-identifier=? #'op o)) infixes)) #'(#%%app op X Y)] [(_ f x ...) #'(#%%app f x ...)]))
- Ranges
- (X +-+ Y) => [X,Y]; (X --- Y) => (X,Y); and same for `+--' and `--+'
- Simple implementation as functions
- Constructors
(definfix ((+-+ X Y) n) (<= X n Y)) ; [X,Y] (definfix ((--- X Y) n) (< X n Y)) ; (X,Y) (definfix ((+-- X Y) n) (and (<= X n) (< n Y))) ; [X,Y) (definfix ((--+ X Y) n) (and (< X n) (<= n Y))) ; (X,Y] (definfix ((== X) n) (= X n)) ; [X,X]
- Set operations
(definfix ((∪ . Rs) n) (ormap (λ(p) (p n)) Rs)) (definfix ((∩ . Rs) n) (andmap (λ(p) (p n)) Rs)) (definfix ((∖ R1 R2) n) (and (R1 n) (not (R2 n)))) ; set-minus, not backslash (define ((¬ R) n) (not (R n)))
- Special sets
(define (∅ n) #f) (define (ℜ n) #t)
(define-syntax-rule (try set)
(apply printf "~a => ~a ~a ~a\n" (~s #:width 23 'set) (let ([pred set]) (for/list ([i 3]) (if (pred i) 'Y 'N)))))
(try ((0 --+ 1) ∪ (0 +-- 2))) (try ((0 +-- 2) ∩ (1 --+ 2))) (try ((0 +-- 3) ∖ (0 --- 1))) (try ((0 +-- 3) ∖ (0 +-+ 1))) </lang>
Output:
((0 --+ 1) ∪ (0 +-- 2)) => Y Y N ((0 +-- 2) ∩ (1 --+ 2)) => N N N ((0 +-- 3) ∖ (0 --- 1)) => Y Y Y ((0 +-- 3) ∖ (0 +-+ 1)) => N N Y
Raku
(formerly Perl 6)
<lang perl6>class Iv {
has $.range handles <min max excludes-min excludes-max minmax ACCEPTS>; method empty {
$.min after $.max or $.min === $.max && ($.excludes-min || $.excludes-max)
} multi method Bool() { not self.empty }; method length() { $.max - $.min } method gist() {
($.excludes-min ?? '(' !! '[') ~ $.min ~ ',' ~ $.max ~ ($.excludes-max ?? ')' !! ']');
}
}
class IvSet {
has Iv @.intervals;
sub canon (@i) {
my @new = consolidate(|@i).grep(*.so); @new.sort(*.range.min);
}
method new(@ranges) {
my @iv = canon @ranges.map: { Iv.new(:range($_)) } self.bless(:intervals(@iv));
}
method complement {
my @new; my @old = @!intervals; if not @old { return iv -Inf..Inf; } my $pre; push @old, Inf^..Inf unless @old[*-1].max === Inf; if @old[0].min === -Inf { $pre = @old.shift; } else { $pre = -Inf..^-Inf; } while @old { my $old = @old.shift; my $excludes-min = !$pre.excludes-max; my $excludes-max = !$old.excludes-min; push @new, Range.new($pre.max,$old.min,:$excludes-min,:$excludes-max); $pre = $old; } IvSet.new(@new);
}
method ACCEPTS(IvSet:D $me: $candidate) {
so $.intervals.any.ACCEPTS($candidate);
} method empty { so $.intervals.all.empty } multi method Bool() { not self.empty };
method length() { [+] $.intervals».length } method gist() { join ' ', $.intervals».gist }
}
sub iv(**@ranges) { IvSet.new(@ranges) }
multi infix:<∩> (Iv $a, Iv $b) {
if $a.min ~~ $b or $a.max ~~ $b or $b.min ~~ $a or $b.max ~~ $a {
my $min = $a.range.min max $b.range.min; my $max = $a.range.max min $b.range.max; my $excludes-min = not $min ~~ $a & $b; my $excludes-max = not $max ~~ $a & $b; Iv.new(:range(Range.new($min,$max,:$excludes-min, :$excludes-max)));
}
} multi infix:<∪> (Iv $a, Iv $b) {
my $min = $a.range.min min $b.range.min; my $max = $a.range.max max $b.range.max; my $excludes-min = not $min ~~ $a | $b; my $excludes-max = not $max ~~ $a | $b; Iv.new(:range(Range.new($min,$max,:$excludes-min, :$excludes-max)));
}
multi infix:<∩> (IvSet $ars, IvSet $brs) {
my @overlap; for $ars.intervals -> $a {
for $brs.intervals -> $b { if $a.min ~~ $b or $a.max ~~ $b or $b.min ~~ $a or $b.max ~~ $a { my $min = $a.range.min max $b.range.min; my $max = $a.range.max min $b.range.max; my $excludes-min = not $min ~~ $a & $b; my $excludes-max = not $max ~~ $a & $b; push @overlap, Range.new($min,$max,:$excludes-min, :$excludes-max); } }
} IvSet.new(@overlap)
}
multi infix:<∪> (IvSet $a, IvSet $b) {
iv |$a.intervals».range, |$b.intervals».range;
}
multi consolidate() { () } multi consolidate($this is copy, *@those) {
gather { for consolidate |@those -> $that { if $this ∩ $that { $this ∪= $that } else { take $that } } take $this; }
}
multi infix:<−> (IvSet $a, IvSet $b) { $a ∩ $b.complement }
multi prefix:<−> (IvSet $a) { $a.complement; }
constant ℝ = iv -Inf..Inf;
my $s1 = iv(0^..1) ∪ iv(0..^2); my $s2 = iv(0..^2) ∩ iv(1^..2); my $s3 = iv(0..^3) − iv(0^..^1); my $s4 = iv(0..^3) − iv(0..1) ;
say "\t\t\t\t0\t1\t2"; say "(0, 1] ∪ [0, 2) -> $s1.gist()\t", 0 ~~ $s1,"\t", 1 ~~ $s1,"\t", 2 ~~ $s1; say "[0, 2) ∩ (1, 2] -> $s2.gist()\t", 0 ~~ $s2,"\t", 1 ~~ $s2,"\t", 2 ~~ $s2; say "[0, 3) − (0, 1) -> $s3.gist()\t", 0 ~~ $s3,"\t", 1 ~~ $s3,"\t", 2 ~~ $s3; say "[0, 3) − [0, 1] -> $s4.gist()\t", 0 ~~ $s4,"\t", 1 ~~ $s4,"\t", 2 ~~ $s4;
say ;
say "ℝ is not empty: ", !ℝ.empty; say "[0,3] − ℝ is empty: ", not iv(0..3) − ℝ;
my $A = iv(0..10) ∩
iv |(0..10).map({ $_ - 1/6 .. $_ + 1/6 }).cache;
my $B = iv 0..sqrt(1/6),
|(1..99).map({ sqrt($_-1/6) .. sqrt($_ + 1/6) }), sqrt(100-1/6)..10;
say 'A − A is empty: ', not $A − $A;
say ;
my $C = $A − $B; say "A − B ="; say " ",.gist for $C.intervals; say "Length A − B = ", $C.length;</lang>
- Output:
0 1 2 (0, 1] ∪ [0, 2) -> [0,2) True True False [0, 2) ∩ (1, 2] -> (1,2) False False False [0, 3) − (0, 1) -> [0,0] [1,3) True True True [0, 3) − [0, 1] -> (1,3) False False True ℝ is not empty: True [0,3] − ℝ is empty: True A − A is empty: True A − B = [0.833333,0.912870929175277) (1.08012344973464,1.166667] [1.833333,1.95789002074512) (2.04124145231932,2.166667] (2.85773803324704,2.97209241668783) (3.02765035409749,3.13581462037113) [3.833333,3.85140666943045) (3.89444048184931,3.97911212877111) (4.02077936060494,4.10284454169706) (4.14326763155202,4.166667] [4.833333,4.88193950529227) (4.91596040125088,4.98330546257535) (5.01663898109747,5.08265022732563) (5.11533641774094,5.166667] (5.84522597225006,5.90197706987526) (5.93014895821906,5.98609499868932) (6.01387285088957,6.06904715201104) (6.09644705272396,6.15088069574865) [6.833333,6.84348838921594) (6.8677992593455,6.91616464041548) (6.94022093788567,6.98808509774554) (7.01189465598754,7.05927286151579) (7.08284312029193,7.12974987873581) (7.15308791129165,7.166667] [7.833333,7.86341740805697) (7.88458411500991,7.92674796706274) (7.94774601171091,7.98957654280459) (8.01040989379861,8.05191488612077) (8.07258735887489,8.11377429642539) (8.13428956127495,8.166667] (8.8411914732499,8.87881373457813) (8.89756521002609,8.93495010245347) (8.95358401237553,8.99073597284078) (9.00925450115972,9.04617783007461) (9.06458309392477,9.10128196098403) (9.1195760135363,9.15605446321358) [9.833333,9.84039294608367) (9.85731538841416,9.89107341663853) (9.9079092984679,9.94149552800449) (9.9582461641931,9.99166319154791) Length A − B = 2.07586484118467
REXX
no error checking, no ∞
<lang rexx>/*REXX program demonstrates a way to represent any set of real numbers and usage. */ call quertySet 1, 3, '[1,2)' call quertySet , , '[0,2) union (1,3)' call quertySet , , '[0,1) union (2,3]' call quertySet , , '[0,2] inter (1,3)' call quertySet , , '(1,2) ∩ (2,3]' call quertySet , , '[0,2) \ (1,3)' say; say center(' start of required tasks ', 40, "═") call quertySet , , '(0,1] union [0,2)' call quertySet , , '[0,2) ∩ (1,3)' call quertySet , , '[0,3] - (0,1)' call quertySet , , '[0,3] - [0,1]' exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ emptySet: parse arg _; nam= valSet(_, 00); return @.3>@.4 /*──────────────────────────────────────────────────────────────────────────────────────*/ isInSet: parse arg #,x; call valSet x
if \datatype(#, 'N') then call set_bad "number isn't not numeric:" # if (@.1=='(' & #<=@.2) |, (@.1=='[' & #< @.2) |, (@.4==')' & #>=@.3) |, (@.4==']' & #> @.3) then return 0 return 1
/*──────────────────────────────────────────────────────────────────────────────────────*/ quertySet: parse arg lv,hv,s1 oop s2 .; op=oop; upper op; cop=
if lv== then lv=0; if hv=="" then hv= 2; if op== then cop= 0 if wordpos(op, '| or UNION') \==0 then cop= "|" if wordpos(op, '& ∩ AND INTER INTERSECTION') \==0 then cop= "&" if wordpos(op, '\ - DIF DIFF DIFFERENCE') \==0 then cop= "\" say do i=lv to hv; b = isInSet(i, s1) if cop\==0 then do b2= isInSet(i, s2) if cop=='&' then b= b & b2 if cop=='|' then b= b | b2 if cop=='\' then b= b & \b2 end express = s1 center(oop, max(5, length(oop) ) ) s2 say right(i, 5) ' is in set' express": " word('no yes', b+1) end /*i*/ return
/*──────────────────────────────────────────────────────────────────────────────────────*/ valSet: parse arg q; q=space(q, 0); L=length(q); @.0= ','; @.4= right(q,1)
parse var q @.1 2 @.2 ',' @.3 (@.4) if @.2>@.3 then parse var L . @.0 @.2 @.3 return space(@.1 @.2 @.0 @.3 @.4, 0)</lang>
- output is the same as the next REXX version (below).
has error checking, ∞ support
<lang rexx>/*REXX program demonstrates a way to represent any set of real numbers and usage. */ call quertySet 1, 3, '[1,2)' call quertySet , , '[0,2) union (1,3)' call quertySet , , '[0,1) union (2,3]' call quertySet , , '[0,2] inter (1,3)' call quertySet , , '(1,2) ∩ (2,3]' call quertySet , , '[0,2) \ (1,3)' say; say center(' start of required tasks ', 40, "═") call quertySet , , '(0,1] union [0,2)' call quertySet , , '[0,2) ∩ (1,3)' call quertySet , , '[0,3] - (0,1)' call quertySet , , '[0,3] - [0,1]' exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ badSet: say; say '***error*** bad format of SET_def: ('arg(1)")"; exit /*──────────────────────────────────────────────────────────────────────────────────────*/ emptySet: parse arg _; nam= valSet(_, 00); return @.3>@.4 /*──────────────────────────────────────────────────────────────────────────────────────*/ isInSet: parse arg #,x; call valSet x
if \datatype(#, 'N') then call set_bad "number isn't not numeric:" # if (@.1=='(' & #<=@.2) |, (@.1=='[' & #< @.2) |, (@.4==')' & #>=@.3) |, (@.4==']' & #> @.3) then return 0 return 1
/*──────────────────────────────────────────────────────────────────────────────────────*/ quertySet: parse arg lv,hv,s1 oop s2 .; op=oop; upper op; cop=
if lv== then lv=0; if hv=="" then hv= 2; if op== then cop= 0 if wordpos(op, '| or UNION') \==0 then cop= "|" if wordpos(op, '& ∩ AND INTER INTERSECTION') \==0 then cop= "&" if wordpos(op, '\ - DIF DIFF DIFFERENCE') \==0 then cop= "\" say do i=lv to hv; b = isInSet(i, s1) if cop\==0 then do b2= isInSet(i, s2) if cop=='&' then b= b & b2 if cop=='|' then b= b | b2 if cop=='\' then b= b & \b2 end express = s1 center(oop, max(5, length(oop) ) ) s2 say right(i, 5) ' is in set' express": " word('no yes', b+1) end /*i*/ return
/*──────────────────────────────────────────────────────────────────────────────────────*/ valSet: parse arg q; q=space(q, 0); L= length(q); @.0= ','
infinity = copies(9, digits() - 1)'e'copies(9, digits() - 1)0 if L<2 then call set_bad 'invalid expression' @.4= right(q, 1) parse var q @.1 2 @.2 ',' @.3 (@.4) if @.1\=='(' & @.1\=="[" then call set_bad 'left boundry' if @.4\==')' & @.4\=="]" then call set_bad 'right boundry' do j=2 to 3; u=@.j; upper u if right(@.j, 1)=='∞' | u="INFINITY" then @.j= '-'infinity if \datatype(@.j, 'N') then call set_bad "value not numeric:" @.j end /*j*/ if @.2>@.3 then parse var L . @.0 @.2 @.3 return space(@.1 @.2 @.0 @.3 @.4, 0)</lang>
- output when using the (internal) default inputs:
1 is in set [1,2) : yes 2 is in set [1,2) : no 3 is in set [1,2) : no 0 is in set [0,2) union (1,3): yes 1 is in set [0,2) union (1,3): yes 2 is in set [0,2) union (1,3): yes 0 is in set [0,1) union (2,3]: yes 1 is in set [0,1) union (2,3]: no 2 is in set [0,1) union (2,3]: no 0 is in set [0,2] inter (1,3): no 1 is in set [0,2] inter (1,3): no 2 is in set [0,2] inter (1,3): yes 0 is in set (1,2) ∩ (2,3]: no 1 is in set (1,2) ∩ (2,3]: no 2 is in set (1,2) ∩ (2,3]: no 0 is in set [0,2) \ (1,3): yes 1 is in set [0,2) \ (1,3): yes 2 is in set [0,2) \ (1,3): no ═══════ start of required tasks ════════ 0 is in set (0,1] union [0,2): yes 1 is in set (0,1] union [0,2): yes 2 is in set (0,1] union [0,2): no 0 is in set [0,2) ∩ (1,3): no 1 is in set [0,2) ∩ (1,3): no 2 is in set [0,2) ∩ (1,3): no 0 is in set [0,3] - (0,1): yes 1 is in set [0,3] - (0,1): yes 2 is in set [0,3] - (0,1): yes 0 is in set [0,3] - [0,1]: no 1 is in set [0,3] - [0,1]: no 2 is in set [0,3] - [0,1]: yes
Ruby
<lang ruby>class Rset
Set = Struct.new(:lo, :hi, :inc_lo, :inc_hi) do def include?(x) (inc_lo ? lo<=x : lo<x) and (inc_hi ? x<=hi : x<hi) end def length hi - lo end def to_s "#{inc_lo ? '[' : '('}#{lo},#{hi}#{inc_hi ? ']' : ')'}" end end def initialize(lo=nil, hi=nil, inc_lo=false, inc_hi=false) if lo.nil? and hi.nil? @sets = [] # empty set else raise TypeError unless lo.is_a?(Numeric) and hi.is_a?(Numeric) raise ArgumentError unless valid?(lo, hi, inc_lo, inc_hi) @sets = [Set[lo, hi, !!inc_lo, !!inc_hi]] # !! -> Boolean values end end def self.[](lo, hi, inc_hi=true) self.new(lo, hi, true, inc_hi) end def self.parse(str) raise ArgumentError unless str =~ /(\[|\()(.+),(.+)(\]|\))/ b0, lo, hi, b1 = $~.captures # $~ : Regexp.last_match lo = Rational(lo) lo = lo.numerator if lo.denominator == 1 hi = Rational(hi) hi = hi.numerator if hi.denominator == 1 self.new(lo, hi, b0=='[', b1==']') end def initialize_copy(obj) super @sets = @sets.map(&:dup) end def include?(x) @sets.any?{|set| set.include?(x)} end def empty? @sets.empty? end def union(other) sets = (@sets+other.sets).map(&:dup).sort_by{|set| [set.lo, set.hi]} work = [] pre = sets.shift sets.each do |post| if valid?(pre.hi, post.lo, !pre.inc_hi, !post.inc_lo) work << pre pre = post else pre.inc_lo |= post.inc_lo if pre.lo == post.lo if pre.hi < post.hi pre.hi = post.hi pre.inc_hi = post.inc_hi elsif pre.hi == post.hi pre.inc_hi |= post.inc_hi end end end work << pre if pre new_Rset(work) end alias | union def intersection(other) sets = @sets.map(&:dup) work = [] other.sets.each do |oset| sets.each do |set| if set.hi < oset.lo or oset.hi < set.lo # ignore elsif oset.lo < set.lo and set.hi < oset.hi work << set else lo = [set.lo, oset.lo].max if set.lo == oset.lo inc_lo = set.inc_lo && oset.inc_lo else inc_lo = (set.lo < oset.lo) ? oset.inc_lo : set.inc_lo end hi = [set.hi, oset.hi].min if set.hi == oset.hi inc_hi = set.inc_hi && oset.inc_hi else inc_hi = (set.hi < oset.hi) ? set.inc_hi : oset.inc_hi end work << Set[lo, hi, inc_lo, inc_hi] if valid?(lo, hi, inc_lo, inc_hi) end end end new_Rset(work) end alias & intersection def difference(other) sets = @sets.map(&:dup) other.sets.each do |oset| work = [] sets.each do |set| if set.hi < oset.lo or oset.hi < set.lo work << set elsif oset.lo < set.lo and set.hi < oset.hi # delete else if set.lo < oset.lo inc_hi = (set.hi==oset.lo and !set.inc_hi) ? false : !oset.inc_lo work << Set[set.lo, oset.lo, set.inc_lo, inc_hi] elsif valid?(set.lo, oset.lo, set.inc_lo, !oset.inc_lo) work << Set[set.lo, set.lo, true, true] end if oset.hi < set.hi inc_lo = (oset.hi==set.lo and !set.inc_lo) ? false : !oset.inc_hi work << Set[oset.hi, set.hi, inc_lo, set.inc_hi] elsif valid?(oset.hi, set.hi, !oset.inc_hi, set.inc_hi) work << Set[set.hi, set.hi, true, true] end end end sets = work end new_Rset(sets) end alias - difference # symmetric difference def ^(other) (self - other) | (other - self) end def ==(other) self.class == other.class and @sets == other.sets end def length @sets.inject(0){|len, set| len + set.length} end def to_s "#{self.class}#{@sets.join}" end alias inspect to_s protected attr_accessor :sets private def new_Rset(sets) rset = self.class.new # empty set rset.sets = sets rset end def valid?(lo, hi, inc_lo, inc_hi) lo < hi or (lo==hi and inc_lo and inc_hi) end
end
def Rset(lo, hi, inc_hi=false)
Rset.new(lo, hi, false, inc_hi)
end</lang>
Test case: <lang ruby>p a = Rset[1,2,false] [1,2,3].each{|x|puts "#{x} => #{a.include?(x)}"} puts a = Rset[0,2,false] #=> Rset[0,2) b = Rset(1,3) #=> Rset(1,3) c = Rset[0,1,false] #=> Rset[0,1) d = Rset(2,3,true) #=> Rset(2,3] puts "#{a} | #{b} -> #{a | b}" puts "#{c} | #{d} -> #{c | d}" puts puts "#{a} & #{b} -> #{a & b}" puts "#{c} & #{d} -> #{c & d}" puts "(#{c} & #{d}).empty? -> #{(c&d).empty?}" puts puts "#{a} - #{b} -> #{a - b}" puts "#{a} - #{a} -> #{a - a}" e = Rset(0,3,true) f = Rset[1,2] puts "#{e} - #{f} -> #{e - f}"
puts "\nTest :" test_set = [["(0, 1]", "|", "[0, 2)"],
["[0, 2)", "&", "(1, 2]"], ["[0, 3)", "-", "(0, 1)"], ["[0, 3)", "-", "[0, 1]"] ]
test_set.each do |sa,ope,sb|
str = "#{sa} #{ope} #{sb}" e = eval("Rset.parse(sa) #{ope} Rset.parse(sb)") puts "%s -> %s" % [str, e] (0..2).each{|i| puts " #{i} : #{e.include?(i)}"}
end
puts test_set = ["x = Rset[0,2] | Rset(3,7) | Rset[8,10]",
"y = Rset(7,9) | Rset(5,6) | Rset[1,4]", "x | y", "x & y", "x - y", "y - x", "x ^ y", "y ^ x == (x | y) - (x & y)"]
x = y = nil test_set.each {|str| puts "#{str} -> #{eval(str)}"}
puts inf = 1.0 / 0.0 # infinity puts "a = #{a = Rset(-inf,inf)}" puts "b = #{b = Rset.parse('[1/3,11/7)')}" puts "a - b -> #{a - b}"</lang>
- Output:
Rset[1,2) 1 => true 2 => false 3 => false Rset[0,2) | Rset(1,3) -> Rset[0,3) Rset[0,1) | Rset(2,3] -> Rset[0,1)(2,3] Rset[0,2) & Rset(1,3) -> Rset(1,2) Rset[0,1) & Rset(2,3] -> Rset (Rset[0,1) & Rset(2,3]).empty? -> true Rset[0,2) - Rset(1,3) -> Rset[0,1] Rset[0,2) - Rset[0,2) -> Rset Rset(0,3] - Rset[1,2] -> Rset(0,1)(2,3] Test : (0, 1] | [0, 2) -> Rset[0,2) 0 : true 1 : true 2 : false [0, 2) & (1, 2] -> Rset(1,2) 0 : false 1 : false 2 : false [0, 3) - (0, 1) -> Rset[0,0][1,3) 0 : true 1 : true 2 : true [0, 3) - [0, 1] -> Rset(1,3) 0 : false 1 : false 2 : true x = Rset[0,2] | Rset(3,7) | Rset[8,10] -> Rset[0,2](3,7)[8,10] y = Rset(7,9) | Rset(5,6) | Rset[1,4] -> Rset[1,4](5,6)(7,9) x | y -> Rset[0,7)(7,10] x & y -> Rset[1,2](3,4](5,6)[8,9) x - y -> Rset[0,1)(4,5][6,7)[9,10] y - x -> Rset(2,3](7,8) x ^ y -> Rset[0,1)(2,3](4,5][6,7)(7,8)[9,10] y ^ x == (x | y) - (x & y) -> true a = Rset(-Infinity,Infinity) b = Rset[1/3,11/7) a - b -> Rset(-Infinity,1/3)[11/7,Infinity)
Optional work:
(with Rational suffix.) <lang ruby>str, e = "e = Rset.new", nil puts "#{str} -> #{eval(str)}\t\t# create empty set" str = "e.empty?" puts "#{str} -> #{eval(str)}" puts
include Math lohi = Enumerator.new do |y|
t = 1 / sqrt(6) 0.step do |n| y << [sqrt(12*n+1) * t, sqrt(12*n+5) * t] y << [sqrt(12*n+7) * t, sqrt(12*n+11) * t] end
end
a = Rset.new loop do
lo, hi = lohi.next break if 10 <= lo a |= Rset(lo, hi)
end a &= Rset(0,10)
b = (0...10).inject(Rset.new){|res,i| res |= Rset(i+1/6r,i+5/6r)}
puts "a : #{a}" puts "a.length : #{a.length}" puts "b : #{b}" puts "b.length : #{b.length}" puts "a - b : #{a - b}" puts "(a-b).length : #{(a-b).length}"</lang>
- Output:
e = Rset.new -> Rset # create empty set e.empty? -> true a : Rset(0.4082482904638631,0.912870929175277)(1.0801234497346435,1.3540064007726602)(1.4719601443879746,1.6832508230603467) ... (9.907909298467901,9.941495528004495)(9.958246164193106,9.991663191547909) a.length : 6.50103079235655 b : Rset(1/6,5/6)(7/6,11/6)(13/6,17/6)(19/6,23/6)(25/6,29/6)(31/6,35/6)(37/6,41/6)(43/6,47/6)(49/6,53/6)(55/6,59/6) b.length : 20/3 a - b : Rset[5/6,0.912870929175277)(1.0801234497346435,7/6][11/6,1.9578900207451218)(2.041241452319315,13/6] ... (9.907909298467901,9.941495528004495)(9.958246164193106,9.991663191547909) (a-b).length : 2.0758648411846745
Tcl
This code represents each set of real numbers as a collection of ranges, where each range is quad of the two boundary values and whether each of those boundaries is a closed boundary. (Using expressions internally would make the code much shorter, at the cost of being much less tractable when it comes to deriving information like the length of the real line “covered” by the set.) A side-effect of the representation is that the length of the list that represents the set is, after normalization, the number of discrete ranges in the set. <lang tcl>package require Tcl 8.5
proc inRange {x range} {
lassign $range a aClosed b bClosed expr {($aClosed ? $a<=$x : $a<$x) && ($bClosed ? $x<=$b : $x<$b)}
} proc normalize {A} {
set A [lsort -index 0 -real [lsort -index 1 -integer -decreasing $A]] for {set i 0} {$i < [llength $A]} {incr i} {
lassign [lindex $A $i] a aClosed b bClosed if {$b < $a || ($a == $b && !($aClosed && $bClosed))} { set A [lreplace $A $i $i] incr i -1 }
} for {set i 0} {$i < [llength $A]} {incr i} {
for {set j [expr {$i+1}]} {$j < [llength $A]} {incr j} { set R [lindex $A $i] lassign [lindex $A $j] a aClosed b bClosed if {[inRange $a $R]} { if {![inRange $b $R]} { lset A $i 2 $b lset A $i 3 $bClosed } set A [lreplace $A $j $j] incr j -1 } }
} return $A
}
proc realset {args} {
set RE {^\s*([\[(])\s*([-\d.e]+|-inf)\s*,\s*([-\d.e]+|inf)\s*([\])])\s*$} set result {} foreach s $args {
if { [regexp $RE $s --> left a b right] && [string is double $a] && [string is double $b] } then { lappend result [list \ $a [expr {$left eq "\["}] $b [expr {$right eq "\]"}]] } else { error "bad range descriptor" }
} return $result
} proc elementOf {x A} {
foreach range $A {
if {[inRange $x $range]} {return 1}
} return 0
} proc union {A B} {
return [normalize [concat $A $B]]
} proc intersection {A B} {
set B [normalize $B] set C {} foreach RA [normalize $A] {
lassign $RA Aa AaClosed Ab AbClosed foreach RB $B { lassign $RB Ba BaClosed Bb BbClosed if {$Aa > $Bb || $Ba > $Ab} continue set RC {} lappend RC [expr {max($Aa,$Ba)}] if {$Aa==$Ba} { lappend RC [expr {min($AaClosed,$BaClosed)}] } else { lappend RC [expr {$Aa>$Ba ? $AaClosed : $BaClosed}] } lappend RC [expr {min($Ab,$Bb)}] if {$Ab==$Bb} { lappend RC [expr {min($AbClosed,$BbClosed)}] } else { lappend RC [expr {$Ab<$Bb ? $AbClosed : $BbClosed}] } lappend C $RC }
} return [normalize $C]
} proc difference {A B} {
set C {} set B [normalize $B] foreach arange [normalize $A] {
if {[isEmpty [intersection [list $arange] $B]]} { lappend C $arange continue } lassign $arange Aa AaClosed Ab AbClosed foreach brange $B { lassign $brange Ba BaClosed Bb BbClosed if {$Bb < $Aa || ($Bb==$Aa && !($AaClosed && $BbClosed))} { continue } if {$Ab < $Ba || ($Ab==$Ba && !($BaClosed && $AbClosed))} { lappend C [list $Aa $AaClosed $Ab $AbClosed] unset arange break } if {$Aa==$Bb} { set AaClosed 0 continue } elseif {$Ab==$Ba} { set AbClosed 0 lappend C [list $Aa $AaClosed $Ab $AbClosed] unset arange continue } if {$Aa<$Ba} { lappend C [list $Aa $AaClosed $Ba [expr {!$BaClosed}]] if {$Ab>$Bb} { set Aa $Bb set AaClosed [expr {!$BbClosed}] } else { unset arange break } } elseif {$Aa==$Ba} { lappend C [list $Aa $AaClosed $Ba [expr {!$BaClosed}]] set Aa $Bb set AaClosed [expr {!$BbClosed}] } else { set Aa $Bb set AaClosed [expr {!$BbClosed}] } } if {[info exist arange]} { lappend C [list $Aa $AaClosed $Ab $AbClosed] }
} return [normalize $C]
} proc isEmpty A {
expr {![llength [normalize $A]]}
} proc length A {
set len 0.0 foreach range [normalize $A] {
lassign $range a _ b _ set len [expr {$len + ($b-$a)}]
} return $len
}</lang> Basic problems: <lang tcl>foreach {str Set} {
{(0, 1] ∪ [0, 2)} {
union [realset {(0,1]}] [realset {[0,2)}]
} {[0, 2) ∩ (1, 2]} {
intersection [realset {[0,2)}] [realset {(1,2]}]
} {[0, 3) − (0, 1)} {
difference [realset {[0,3)}] [realset {(0,1)}]
} {[0, 3) − [0, 1]} {
difference [realset {[0,3)}] [realset {[0,1]}]
}
} {
set Set [eval $Set] foreach x {0 1 2} {
puts "$x : $str :\t[elementOf $x $Set]"
}
}</lang> Extra credit: <lang tcl>proc spi2 {from to} {
for {set i $from} {$i<=$to} {incr i} {
lappend result [list [expr {$i+1./6}] 0 [expr {$i+5./6}] 0]
} return [intersection [list [list $from 0 $to 0]] $result]
} proc applyfunc {var func} {
upvar 1 $var A for {set i 0} {$i < [llength $A]} {incr i} {
lassign [lindex $A $i] a - b - lset A $i 0 [$func $a] lset A $i 2 [$func $b]
}
} set A [spi2 0 100] applyfunc A ::tcl::mathfunc::sqrt set B [spi2 0 10] set AB [difference $A $B] puts "[llength $AB] contiguous subsets, total length [length $AB]"</lang> Output:
0 : (0, 1] ∪ [0, 2) : 1 1 : (0, 1] ∪ [0, 2) : 1 2 : (0, 1] ∪ [0, 2) : 0 0 : [0, 2) ∩ (1, 2] : 0 1 : [0, 2) ∩ (1, 2] : 0 2 : [0, 2) ∩ (1, 2] : 0 0 : [0, 3) − (0, 1) : 1 1 : [0, 3) − (0, 1) : 1 2 : [0, 3) − (0, 1) : 1 0 : [0, 3) − [0, 1] : 0 1 : [0, 3) − [0, 1] : 0 2 : [0, 3) − [0, 1] : 1 40 contiguous subsets, total length 2.075864841184667
Wren
<lang ecmascript>import "/dynamic" for Enum import "/math" for Math
var RangeType = Enum.create("RangeType", ["CLOSED", "BOTH_OPEN", "LEFT_OPEN", "RIGHT_OPEN"])
class RealSet {
construct new(start, end, pred) { _low = start _high = end _pred = (pred == RangeType.CLOSED) ? Fn.new { |d| d >= _low && d <= _high } : (pred == RangeType.BOTH_OPEN) ? Fn.new { |d| d > _low && d < _high } : (pred == RangeType.LEFT_OPEN) ? Fn.new { |d| d > _low && d <= _high } : (pred == RangeType.RIGHT_OPEN) ? Fn.new { |d| d >= _low && d < _high } : pred }
low { _low } high { _high } pred { _pred }
contains(d) { _pred.call(d) }
union(other) { if (!other.type == RealSet) Fiber.abort("Argument must be a RealSet") var low2 = Math.min(_low, other.low) var high2 = Math.max(_high, other.high) return RealSet.new(low2, high2) { |d| _pred.call(d) || other.pred.call(d) } }
intersect(other) { if (!other.type == RealSet) Fiber.abort("Argument must be a RealSet") var low2 = Math.max(_low, other.low) var high2 = Math.min(_high, other.high) return RealSet.new(low2, high2) { |d| _pred.call(d) && other.pred.call(d) } }
subtract(other) { if (!other.type == RealSet) Fiber.abort("Argument must be a RealSet") return RealSet.new(_low, _high) { |d| _pred.call(d) && !other.pred.call(d) } }
length { if (_low.isInfinity || _high.isInfinity) return -1 // error value if (_high <= _low) return 0 var p = _low var count = 0 var interval = 0.00001 while (true) { if (_pred.call(p)) count = count + 1 p = p + interval if (p >= _high) break } return count * interval }
isEmpty { (_high == _low) ? !_pred.call(_low) : length == 0 }
}
var a = RealSet.new(0, 1, RangeType.LEFT_OPEN) var b = RealSet.new(0, 2, RangeType.RIGHT_OPEN) var c = RealSet.new(1, 2, RangeType.LEFT_OPEN) var d = RealSet.new(0, 3, RangeType.RIGHT_OPEN) var e = RealSet.new(0, 1, RangeType.BOTH_OPEN) var f = RealSet.new(0, 1, RangeType.CLOSED) var g = RealSet.new(0, 0, RangeType.CLOSED)
for (i in 0..2) {
System.print("(0, 1] ∪ [0, 2) contains %(i) is %(a.union(b).contains(i))") System.print("[0, 2) ∩ (1, 2] contains %(i) is %(b.intersect(c).contains(i))") System.print("[0, 3) − (0, 1) contains %(i) is %(d.subtract(e).contains(i))") System.print("[0, 3) − [0, 1] contains %(i) is %(d.subtract(f).contains(i))\n")
}
System.print("[0, 0] is empty is %(g.isEmpty)\n")
var aa = RealSet.new(0, 10) { |x| (0 < x && x < 10) && ((Num.pi * x * x).sin.abs > 0.5) } var bb = RealSet.new(0, 10) { |x| (0 < x && x < 10) && ((Num.pi * x).sin.abs > 0.5) } var cc = aa.subtract(bb) System.print("Approx length of A - B is %(cc.length)")</lang>
- Output:
(0, 1] ∪ [0, 2) contains 0 is true [0, 2) ∩ (1, 2] contains 0 is false [0, 3) − (0, 1) contains 0 is true [0, 3) − [0, 1] contains 0 is false (0, 1] ∪ [0, 2) contains 1 is true [0, 2) ∩ (1, 2] contains 1 is false [0, 3) − (0, 1) contains 1 is true [0, 3) − [0, 1] contains 1 is false (0, 1] ∪ [0, 2) contains 2 is false [0, 2) ∩ (1, 2] contains 2 is false [0, 3) − (0, 1) contains 2 is true [0, 3) − [0, 1] contains 2 is true [0, 0] is empty is false Approx length of A - B is 2.07587
zkl
No ∞ <lang zkl>class RealSet{
fcn init(fx){ var [const] contains=fx; } fcn holds(x){ contains(x) } fcn __opAdd(rs){ RealSet('wrap(x){ contains(x) or rs.contains(x) }) } fcn __opSub(rs){ RealSet('wrap(x){ contains(x) and not rs.contains(x) }) } fcn intersection(rs) { RealSet('wrap(x){ contains(x) and rs.contains(x) }) }
}</lang> The python method could used but the zkl compiler is slow when used in code to generate code.
The method used is a bit inefficient because it closes the contains function of the other set so you can build quite a long call chain as you create new sets. <lang zkl>tester := TheVault.Test.UnitTester.UnitTester();
// test union
s:=RealSet(fcn(x){ 0.0 < x <= 1.0 }) +
RealSet(fcn(x){ 0.0 <= x < 1.0 });
tester.testRun(s.holds(0.0),Void,True,__LINE__); tester.testRun(s.holds(1.0),Void,True,__LINE__); tester.testRun(s.holds(2.0),Void,False,__LINE__);
// test difference
s1 := RealSet(fcn(x){ 0.0 <= x < 3.0 }) -
RealSet(fcn(x){ 0.0 < x < 1.0 });
tester.testRun(s1.holds(0.0),Void,True,__LINE__); tester.testRun(s1.holds(0.5),Void,False,__LINE__); tester.testRun(s1.holds(1.0),Void,True,__LINE__); tester.testRun(s1.holds(2.0),Void,True,__LINE__);
s2 := RealSet(fcn(x){ 0.0 <= x < 3.0 }) -
RealSet(fcn(x){ 0.0 <= x <= 1.0 });
tester.testRun(s2.holds(0.0),Void,False,__LINE__); tester.testRun(s2.holds(1.0),Void,False,__LINE__); tester.testRun(s2.holds(2.0),Void,True,__LINE__);
// test intersection
s := RealSet(fcn(x){ 0.0 <= x < 2.0 }).intersection(
RealSet(fcn(x){ 1.0 < x <= 2.0 }));
tester.testRun(s.holds(0.0),Void,False,__LINE__); tester.testRun(s.holds(1.0),Void,False,__LINE__); tester.testRun(s.holds(2.0),Void,False,__LINE__);</lang>
- Output:
$ zkl bbb ===================== Unit Test 1 ===================== Test 1 passed! ===================== Unit Test 2 ===================== Test 2 passed! ... ===================== Unit Test 12 ===================== Test 12 passed! ===================== Unit Test 13 ===================== Test 13 passed!